image-match/python/py/Lib/site-packages/scipy/stats/tests/test_stats.py

""" Test functions for stats module

    WRITTEN BY LOUIS LUANGKESORN <lluang@yahoo.com> FOR THE STATS MODULE
    BASED ON WILKINSON'S STATISTICS QUIZ
    https://www.stanford.edu/~clint/bench/wilk.txt

    Additional tests by a host of SciPy developers.
"""
import math
import os
import re
import warnings
from collections import namedtuple
from itertools import product
import hypothesis.extra.numpy as npst
import hypothesis
import contextlib

from numpy.testing import (assert_, assert_equal,
                           assert_almost_equal, assert_array_almost_equal,
                           assert_array_equal, assert_approx_equal,
                           assert_allclose, assert_array_less)
import pytest
from pytest import raises as assert_raises
from numpy import array, arange, float32, power
import numpy as np

import scipy.stats as stats
import scipy.stats._mstats_basic as mstats_basic
from scipy.stats._ksstats import kolmogn
from scipy.special._testutils import FuncData
from scipy import optimize, special
from .common_tests import check_named_results
from scipy.stats._axis_nan_policy import (_broadcast_concatenate, SmallSampleWarning,
                                          too_small_nd_omit, too_small_nd_not_omit,
                                          too_small_1d_omit, too_small_1d_not_omit)
from scipy.stats._stats_py import (_chk_asarray, _moment,
                                   LinregressResult, _xp_mean, _xp_var, _SimpleChi2)
from scipy._lib._util import AxisError
from scipy.conftest import skip_xp_invalid_arg
from scipy._lib._array_api import (array_namespace, eager_warns, is_lazy_array,
                                   is_numpy, is_torch, xp_default_dtype, xp_size,
                                   SCIPY_ARRAY_API, make_xp_test_case, xp_ravel,
                                   xp_swapaxes)
from scipy._lib._array_api_no_0d import xp_assert_close, xp_assert_equal
import scipy._lib.array_api_extra as xpx

lazy_xp_modules = [stats]
skip_xp_backends = pytest.mark.skip_xp_backends
xfail_xp_backends = pytest.mark.xfail_xp_backends

""" Numbers in docstrings beginning with 'W' refer to the section numbers
    and headings found in the STATISTICS QUIZ of Leland Wilkinson.  These are
    considered to be essential functionality.  True testing and
    evaluation of a statistics package requires use of the
    NIST Statistical test data.  See McCoullough(1999) Assessing The Reliability
    of Statistical Software for a test methodology and its
    implementation in testing SAS, SPSS, and S-Plus
"""

#  Datasets
#  These data sets are from the nasty.dat sets used by Wilkinson
#  For completeness, I should write the relevant tests and count them as failures
#  Somewhat acceptable, since this is still beta software.  It would count as a
#  good target for 1.0 status
X = array([1,2,3,4,5,6,7,8,9], float)
ZERO = array([0,0,0,0,0,0,0,0,0], float)
BIG = array([99999991,99999992,99999993,99999994,99999995,99999996,99999997,
             99999998,99999999], float)
LITTLE = array([0.99999991,0.99999992,0.99999993,0.99999994,0.99999995,0.99999996,
                0.99999997,0.99999998,0.99999999], float)
HUGE = array([1e+12,2e+12,3e+12,4e+12,5e+12,6e+12,7e+12,8e+12,9e+12], float)
TINY = array([1e-12,2e-12,3e-12,4e-12,5e-12,6e-12,7e-12,8e-12,9e-12], float)
ROUND = array([0.5,1.5,2.5,3.5,4.5,5.5,6.5,7.5,8.5], float)


class TestTrimmedStats:
    # TODO: write these tests to handle missing values properly
    dprec = np.finfo(np.float64).precision

    @make_xp_test_case(stats.tmean)
    def test_tmean(self, xp):
        default_dtype = xp_default_dtype(xp)
        x = xp.asarray(X, dtype=default_dtype)

        y = stats.tmean(x, (2, 8), (True, True))
        xp_assert_close(y, xp.asarray(5.0))

        y1 = stats.tmean(x, limits=(2, 8), inclusive=(False, False))
        y2 = stats.tmean(x, limits=None)
        xp_assert_close(y1, y2)

        x_2d = xp.reshape(xp.arange(63.), (9, 7))
        y = stats.tmean(x_2d, axis=None)
        xp_assert_close(y, xp.mean(x_2d))

        y = stats.tmean(x_2d, axis=0)
        xp_assert_close(y, xp.mean(x_2d, axis=0))

        y = stats.tmean(x_2d, axis=1)
        xp_assert_close(y, xp.mean(x_2d, axis=1))

        y = stats.tmean(x_2d, limits=(2, 61), axis=None)
        xp_assert_close(y, xp.asarray(31.5))

        y = stats.tmean(x_2d, limits=(2, 21), axis=0)
        y_true = [14, 11.5, 9, 10, 11, 12, 13]
        xp_assert_close(y, xp.asarray(y_true))

        y = stats.tmean(x_2d, limits=(2, 21), inclusive=(True, False), axis=0)
        y_true = [10.5, 11.5, 9, 10, 11, 12, 13]
        xp_assert_close(y, xp.asarray(y_true))

        x_2d_with_nan = xpx.at(x_2d)[-1, -3:].set(xp.nan, copy=True)
        y = stats.tmean(x_2d_with_nan, limits=(1, 13), axis=0)
        y_true = [7, 4.5, 5.5, 6.5, xp.nan, xp.nan, xp.nan]
        xp_assert_close(y, xp.asarray(y_true))

        y = stats.tmean(x_2d, limits=(2, 21), axis=1)
        y_true = [4, 10, 17, 21, xp.nan, xp.nan, xp.nan, xp.nan, xp.nan]
        xp_assert_close(y, xp.asarray(y_true))

        y = stats.tmean(x_2d, limits=(2, 21),
                        inclusive=(False, True), axis=1)
        y_true = [4.5, 10, 17, 21, xp.nan, xp.nan, xp.nan, xp.nan, xp.nan]
        xp_assert_close(y, xp.asarray(y_true))

    @make_xp_test_case(stats.tvar)
    @pytest.mark.filterwarnings(
        "ignore:invalid value encountered in divide:RuntimeWarning:dask"
    )
    def test_tvar(self, xp):
        x = xp.asarray(X.tolist())  # use default dtype of xp

        y = stats.tvar(x, limits=(2, 8), inclusive=(True, True))
        xp_assert_close(y, xp.asarray(4.6666666666666661))

        y = stats.tvar(x, limits=None)
        xp_assert_close(y, xp.var(x, correction=1))

        x_2d = xp.reshape(xp.arange(63.), (9, 7))
        y = stats.tvar(x_2d, axis=None)
        xp_assert_close(y, xp.var(x_2d, correction=1))

        y = stats.tvar(x_2d, axis=0)
        xp_assert_close(y, xp.full((7,), 367.5))

        y = stats.tvar(x_2d, axis=1)
        xp_assert_close(y, xp.full((9,), 4.66666667))

        # Limiting some values along one axis
        y = stats.tvar(x_2d, limits=(1, 5), axis=1, inclusive=(True, True))
        xp_assert_close(y[0], xp.asarray(2.5))

        # Limiting all values along one axis
        y = stats.tvar(x_2d, limits=(0, 6), axis=1, inclusive=(True, True))
        xp_assert_close(y[0], xp.asarray(4.666666666666667))
        xp_assert_equal(y[1], xp.asarray(xp.nan))

    @make_xp_test_case(stats.tstd)
    def test_tstd(self, xp):
        x = xp.asarray(X.tolist())  # use default dtype of xp

        y = stats.tstd(x, (2, 8), (True, True))
        xp_assert_close(y, xp.asarray(2.1602468994692865))

        y = stats.tstd(x, limits=None)
        xp_assert_close(y, xp.std(x, correction=1))

    @make_xp_test_case(stats.tmin)
    def test_tmin(self, xp):
        x = xp.arange(10.)
        xp_assert_equal(stats.tmin(x), xp.asarray(0.))
        xp_assert_equal(stats.tmin(x, lowerlimit=0), xp.asarray(0.))
        xp_assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), xp.asarray(1.))

        x = xp.reshape(x, (5, 2))
        xp_assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False),
                        xp.asarray([2., 1.]))
        xp_assert_equal(stats.tmin(x, axis=1), xp.asarray([0., 2., 4., 6., 8.]))
        xp_assert_equal(stats.tmin(x, axis=None), xp.asarray(0.))

        x = xpx.at(xp.arange(10.), 9).set(xp.nan)
        xp_assert_equal(stats.tmin(x), xp.asarray(xp.nan))

        # check that if a full slice is masked, the output returns a
        # nan instead of a garbage value.
        x = xp.reshape(xp.arange(16), (4, 4))
        res = stats.tmin(x, lowerlimit=4, axis=1)
        xp_assert_equal(res, xp.asarray([np.nan, 4, 8, 12]))

    @skip_xp_backends(np_only=True,
                      reason="Only NumPy arrays support scalar input/`nan_policy`.")
    def test_tmin_scalar_and_nanpolicy(self, xp):
        assert_equal(stats.tmin(4), 4)

        x = np.arange(10.)
        x[9] = np.nan
        with warnings.catch_warnings():
            warnings.filterwarnings("ignore", "invalid value", RuntimeWarning)
            assert_equal(stats.tmin(x, nan_policy='omit'), 0.)
            msg = "The input contains nan values"
            with assert_raises(ValueError, match=msg):
                stats.tmin(x, nan_policy='raise')
            msg = "nan_policy must be one of..."
            with assert_raises(ValueError, match=msg):
                stats.tmin(x, nan_policy='foobar')

    @make_xp_test_case(stats.tmax)
    def test_tmax(self, xp):
        x = xp.arange(10.)
        xp_assert_equal(stats.tmax(x), xp.asarray(9.))
        xp_assert_equal(stats.tmax(x, upperlimit=9), xp.asarray(9.))
        xp_assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), xp.asarray(8.))

        x = xp.reshape(x, (5, 2))
        xp_assert_equal(stats.tmax(x, upperlimit=9, inclusive=False),
                        xp.asarray([8., 7.]))
        xp_assert_equal(stats.tmax(x, axis=1), xp.asarray([1., 3., 5., 7., 9.]))
        xp_assert_equal(stats.tmax(x, axis=None), xp.asarray(9.))

        x = xpx.at(xp.arange(10.), 9).set(xp.nan)
        xp_assert_equal(stats.tmax(x), xp.asarray(xp.nan))

        # check that if a full slice is masked, the output returns a
        # nan instead of a garbage value.
        with warnings.catch_warnings():
            warnings.filterwarnings(
                "ignore", "All-NaN slice encountered", RuntimeWarning)
            x = xp.reshape(xp.arange(16), (4, 4))
            res = stats.tmax(x, upperlimit=11, axis=1)
            xp_assert_equal(res, xp.asarray([3, 7, 11, np.nan]))

    @skip_xp_backends(np_only=True,
                      reason="Only NumPy arrays support scalar input/`nan_policy`.")
    def test_tmax_scalar_and_nanpolicy(self, xp):
        assert_equal(stats.tmax(4), 4)

        x = np.arange(10.)
        x[6] = np.nan
        with warnings.catch_warnings():
            warnings.filterwarnings("ignore", "invalid value", RuntimeWarning)
            assert_equal(stats.tmax(x, nan_policy='omit'), 9.)
            msg = "The input contains nan values"
            with assert_raises(ValueError, match=msg):
                stats.tmax(x, nan_policy='raise')
            msg = "nan_policy must be one of..."
            with assert_raises(ValueError, match=msg):
                stats.tmax(x, nan_policy='foobar')

    @make_xp_test_case(stats.tmin, stats.tmax)
    def test_tmin_tmax_int_dtype(self, xp):
        x = xp.reshape(xp.arange(10, dtype=xp.int16), (2, 5)).T

        # When tmin/tmax don't need to inject any NaNs,
        # retain the input dtype. Dask/JAX can't inspect
        # the data so they always return float.
        expect_dtype = xp_default_dtype(xp) if is_lazy_array(x) else x.dtype
        xp_assert_equal(stats.tmin(x), xp.asarray([0, 5], dtype=expect_dtype))
        xp_assert_equal(stats.tmax(x), xp.asarray([4, 9], dtype=expect_dtype))

        # When they do inject NaNs, all backends behave the same.
        xp_assert_equal(stats.tmin(x, lowerlimit=6), xp.asarray([xp.nan, 6.]))
        xp_assert_equal(stats.tmax(x, upperlimit=3), xp.asarray([3., xp.nan]))

    @skip_xp_backends(eager_only=True, reason="Only with data-dependent output dtype")
    @make_xp_test_case(stats.tmin, stats.tmax)
    def test_gh_22626(self, xp):
        # Test that `tmin`/`tmax` returns exact result with outrageously large integers
        x = xp.arange(2**62, 2**62+10)
        xp_assert_equal(stats.tmin(x[None, :]), x)
        xp_assert_equal(stats.tmax(x[None, :]), x)

    @make_xp_test_case(stats.tsem)
    def test_tsem(self, xp):
        x = xp.asarray(X.tolist())  # use default dtype of xp

        y = stats.tsem(x, limits=(3, 8), inclusive=(False, True))
        y_ref = xp.asarray([4., 5., 6., 7., 8.])
        xp_assert_close(y, xp.std(y_ref, correction=1) / xp_size(y_ref)**0.5)
        xp_assert_close(stats.tsem(x, limits=[-1, 10]), stats.tsem(x, limits=None))


class TestPearsonrWilkinson:
    """ W.II.D. Compute a correlation matrix on all the variables.

        All the correlations, except for ZERO and MISS, should be exactly 1.
        ZERO and MISS should have undefined or missing correlations with the
        other variables.  The same should go for SPEARMAN correlations, if
        your program has them.
    """

    def test_pXX(self):
        y = stats.pearsonr(X,X)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXBIG(self):
        y = stats.pearsonr(X,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXLITTLE(self):
        y = stats.pearsonr(X,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXHUGE(self):
        y = stats.pearsonr(X,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXTINY(self):
        y = stats.pearsonr(X,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pXROUND(self):
        y = stats.pearsonr(X,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGBIG(self):
        y = stats.pearsonr(BIG,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGLITTLE(self):
        y = stats.pearsonr(BIG,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGHUGE(self):
        y = stats.pearsonr(BIG,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGTINY(self):
        y = stats.pearsonr(BIG,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pBIGROUND(self):
        y = stats.pearsonr(BIG,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLELITTLE(self):
        y = stats.pearsonr(LITTLE,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLEHUGE(self):
        y = stats.pearsonr(LITTLE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLETINY(self):
        y = stats.pearsonr(LITTLE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pLITTLEROUND(self):
        y = stats.pearsonr(LITTLE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pHUGEHUGE(self):
        y = stats.pearsonr(HUGE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pHUGETINY(self):
        y = stats.pearsonr(HUGE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pHUGEROUND(self):
        y = stats.pearsonr(HUGE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pTINYTINY(self):
        y = stats.pearsonr(TINY,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pTINYROUND(self):
        y = stats.pearsonr(TINY,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_pROUNDROUND(self):
        y = stats.pearsonr(ROUND,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)


@make_xp_test_case(stats.pearsonr)
class TestPearsonr:
    def test_pearsonr_result_attributes(self):
        res = stats.pearsonr(X, X)
        attributes = ('correlation', 'pvalue')
        check_named_results(res, attributes)
        assert_equal(res.correlation, res.statistic)

    def test_r_almost_exactly_pos1(self, xp):
        a = xp.arange(3.0)
        r, prob = stats.pearsonr(a, a)

        xp_assert_close(r, xp.asarray(1.0), atol=1e-15)
        # With n = len(a) = 3, the error in prob grows like the
        # square root of the error in r.
        xp_assert_close(prob, xp.asarray(0.0), atol=np.sqrt(2*np.spacing(1.0)))

    def test_r_almost_exactly_neg1(self, xp):
        a = xp.arange(3.0)
        r, prob = stats.pearsonr(a, -a)

        xp_assert_close(r, xp.asarray(-1.0), atol=1e-15)
        # With n = len(a) = 3, the error in prob grows like the
        # square root of the error in r.
        xp_assert_close(prob, xp.asarray(0.0), atol=np.sqrt(2*np.spacing(1.0)))

    def test_basic(self, xp):
        # A basic test, with a correlation coefficient
        # that is not 1 or -1.
        a = xp.asarray([-1, 0, 1])
        b = xp.asarray([0, 0, 3])
        r, prob = stats.pearsonr(a, b)
        xp_assert_close(r, xp.asarray(3**0.5/2))
        xp_assert_close(prob, xp.asarray(1/3))

    def test_constant_input(self, xp):
        # Zero variance input
        # See https://github.com/scipy/scipy/issues/3728
        x = xp.asarray([0.667, 0.667, 0.667])
        y = xp.asarray([0.123, 0.456, 0.789])
        msg = "An input array is constant"
        with eager_warns(stats.ConstantInputWarning, match=msg, xp=xp):
            r, p = stats.pearsonr(x, y)
            xp_assert_close(r, xp.asarray(xp.nan))
            xp_assert_close(p, xp.asarray(xp.nan))

    @pytest.mark.parametrize('dtype', ['float32', 'float64'])
    def test_near_constant_input(self, xp, dtype):
        npdtype = getattr(np, dtype)
        dtype = getattr(xp, dtype)
        # Near constant input (but not constant):
        x = xp.asarray([2, 2, 2 + np.spacing(2, dtype=npdtype)], dtype=dtype)
        y = xp.asarray([3, 3, 3 + 6*np.spacing(3, dtype=npdtype)], dtype=dtype)
        msg = "An input array is nearly constant; the computed"
        with eager_warns(stats.NearConstantInputWarning, match=msg, xp=xp):
            # r and p are garbage, so don't bother checking them in this case.
            # (The exact value of r would be 1.)
            stats.pearsonr(x, y)

    def test_very_small_input_values(self, xp):
        # Very small values in an input.  A naive implementation will
        # suffer from underflow.
        # See https://github.com/scipy/scipy/issues/9353
        x = xp.asarray([0.004434375, 0.004756007, 0.003911996, 0.0038005, 0.003409971],
                       dtype=xp.float64)
        y = xp.asarray([2.48e-188, 7.41e-181, 4.09e-208, 2.08e-223, 2.66e-245],
                       dtype=xp.float64)
        r, p = stats.pearsonr(x, y)

        # The expected values were computed using mpmath with 80 digits
        # of precision.
        xp_assert_close(r, xp.asarray(0.7272930540750450, dtype=xp.float64))
        xp_assert_close(p, xp.asarray(0.1637805429533202, dtype=xp.float64))

    def test_very_large_input_values(self, xp):
        # Very large values in an input.  A naive implementation will
        # suffer from overflow.
        # See https://github.com/scipy/scipy/issues/8980
        x = 1e90*xp.asarray([0, 0, 0, 1, 1, 1, 1], dtype=xp.float64)
        y = 1e90*xp.arange(7, dtype=xp.float64)

        r, p = stats.pearsonr(x, y)

        # The expected values were computed using mpmath with 80 digits
        # of precision.
        xp_assert_close(r, xp.asarray(0.8660254037844386, dtype=xp.float64))
        xp_assert_close(p, xp.asarray(0.011724811003954638, dtype=xp.float64))

    def test_extremely_large_input_values(self, xp):
        # Extremely large values in x and y.  These values would cause the
        # product sigma_x * sigma_y to overflow if the two factors were
        # computed independently.
        x = xp.asarray([2.3e200, 4.5e200, 6.7e200, 8e200], dtype=xp.float64)
        y = xp.asarray([1.2e199, 5.5e200, 3.3e201, 1.0e200], dtype=xp.float64)
        r, p = stats.pearsonr(x, y)

        # The expected values were computed using mpmath with 80 digits
        # of precision.
        xp_assert_close(r, xp.asarray(0.351312332103289, dtype=xp.float64))
        xp_assert_close(p, xp.asarray(0.648687667896711, dtype=xp.float64))

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_length_two_pos1(self, xp):
        # Inputs with length 2.
        # See https://github.com/scipy/scipy/issues/7730
        x = xp.asarray([1., 2.])
        y = xp.asarray([3., 5.])
        res = stats.pearsonr(x, y)
        r, p = res
        one = xp.asarray(1.)
        xp_assert_equal(r, one)
        xp_assert_equal(p, one)
        low, high = res.confidence_interval()
        xp_assert_equal(low, -one)
        xp_assert_equal(high, one)

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_length_two_neg1(self, xp):
        # Inputs with length 2.
        # See https://github.com/scipy/scipy/issues/7730
        x = xp.asarray([2., 1.])
        y = xp.asarray([3., 5.])
        res = stats.pearsonr(x, y)
        r, p = res
        one = xp.asarray(1.)
        xp_assert_equal(r, -one)
        xp_assert_equal(p, one)
        low, high = res.confidence_interval()
        xp_assert_equal(low, -one)
        xp_assert_equal(high, one)

    @pytest.mark.filterwarnings("ignore:invalid value encountered in divide")
    def test_length_two_constant_input(self, xp):
        # Zero variance input
        # See https://github.com/scipy/scipy/issues/3728
        # and https://github.com/scipy/scipy/issues/7730
        x = xp.asarray([0.667, 0.667])
        y = xp.asarray([0.123, 0.456])
        msg = "An input array is constant"
        with eager_warns(stats.ConstantInputWarning, match=msg, xp=xp):
            r, p = stats.pearsonr(x, y)
            xp_assert_close(r, xp.asarray(xp.nan))
            xp_assert_close(p, xp.asarray(xp.nan))

    # Expected values computed with R 3.6.2 cor.test, e.g.
    # options(digits=16)
    # x <- c(1, 2, 3, 4)
    # y <- c(0, 1, 0.5, 1)
    # cor.test(x, y, method = "pearson", alternative = "g")
    # correlation coefficient and p-value for alternative='two-sided'
    # calculated with mpmath agree to 16 digits.
    @skip_xp_backends(np_only=True)
    @pytest.mark.parametrize('alternative, pval, rlow, rhigh, sign',
            [('two-sided', 0.325800137536, -0.814938968841, 0.99230697523, 1),
             ('less', 0.8370999312316, -1, 0.985600937290653, 1),
             ('greater', 0.1629000687684, -0.6785654158217636, 1, 1),
             ('two-sided', 0.325800137536, -0.992306975236, 0.81493896884, -1),
             ('less', 0.1629000687684, -1.0, 0.6785654158217636, -1),
             ('greater', 0.8370999312316, -0.985600937290653, 1.0, -1)])
    def test_basic_example(self, alternative, pval, rlow, rhigh, sign, xp):
        x = [1, 2, 3, 4]
        y = np.array([0, 1, 0.5, 1]) * sign
        result = stats.pearsonr(x, y, alternative=alternative)
        assert_allclose(result.statistic, 0.6741998624632421*sign, rtol=1e-12)
        assert_allclose(result.pvalue, pval, rtol=1e-6)
        ci = result.confidence_interval()
        assert_allclose(ci, (rlow, rhigh), rtol=1e-6)

    def test_negative_correlation_pvalue_gh17795(self, xp):
        x = xp.arange(10.)
        y = -x
        test_greater = stats.pearsonr(x, y, alternative='greater')
        test_less = stats.pearsonr(x, y, alternative='less')
        xp_assert_close(test_greater.pvalue, xp.asarray(1.))
        xp_assert_close(test_less.pvalue, xp.asarray(0.), atol=1e-20)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_length3_r_exactly_negative_one(self, xp):
        x = xp.asarray([1., 2., 3.])
        y = xp.asarray([5., -4., -13.])
        res = stats.pearsonr(x, y)

        # The expected r and p are exact.
        r, p = res
        one = xp.asarray(1.0)
        xp_assert_close(r, -one)
        xp_assert_close(p, 0*one, atol=1e-7)
        low, high = res.confidence_interval()
        xp_assert_equal(low, -one)
        xp_assert_equal(high, one)

    def test_input_validation(self):
        # Arraylike is np only
        x = [1, 2, 3]
        y = [4]
        message = '`x` and `y` must have the same length along `axis`.'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, y)

        x = [1, 2, 3]
        y = [4, 5]
        message = '`x` and `y` must be broadcastable.'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, y)

        x = [1]
        y = [2]
        message = '`x` and `y` must have length at least 2.'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, y)

        x = [-1j, -2j, -3.0j]
        y = [-1j, -2j, -3.0j]
        message = 'This function does not support complex data'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, y)

        message = "`method` must be an instance of..."
        with pytest.raises(ValueError, match=message):
            stats.pearsonr([1, 2], [3, 4], method="asymptotic")

        res = stats.pearsonr([1, 2], [3, 4])
        with pytest.raises(ValueError, match=message):
            res.confidence_interval(method="exact")

    @pytest.mark.fail_slow(10)
    @pytest.mark.xfail_on_32bit("Monte Carlo method needs > a few kB of memory")
    @pytest.mark.parametrize('alternative', ('less', 'greater', 'two-sided'))
    @pytest.mark.parametrize('method_name',
                             ('permutation', 'monte_carlo', 'monte_carlo2'))
    def test_resampling_pvalue(self, method_name, alternative):
        rng = np.random.default_rng(24623935790378923)
        size = (2, 100) if method_name == 'permutation' else (2, 1000)
        x = rng.normal(size=size)
        y = rng.normal(size=size)
        methods = {'permutation': stats.PermutationMethod(rng=rng),
                   'monte_carlo': stats.MonteCarloMethod(rvs=(rng.normal,)*2),
                   'monte_carlo2': stats.MonteCarloMethod(rng=1294)}
        method = methods[method_name]
        res = stats.pearsonr(x, y, alternative=alternative, method=method, axis=-1)
        ref = stats.pearsonr(x, y, alternative=alternative, axis=-1)
        assert_allclose(res.statistic, ref.statistic, rtol=1e-15)
        assert_allclose(res.pvalue, ref.pvalue, rtol=1e-2, atol=1e-3)

        if method_name == 'monte_carlo2':
            method = stats.MonteCarloMethod(rng=1294)
            res2 = stats.pearsonr(x, y, alternative=alternative, method=method, axis=-1)
            assert_equal(res2.statistic, res.statistic)
            assert_equal(res2.pvalue, res.pvalue)

    @pytest.mark.slow
    @pytest.mark.parametrize('alternative', ('less', 'greater', 'two-sided'))
    def test_bootstrap_ci(self, alternative):
        rng = np.random.default_rng(2462935790378923)
        x = rng.normal(size=(2, 100))
        y = rng.normal(size=(2, 100))
        res = stats.pearsonr(x, y, alternative=alternative, axis=-1)

        # preserve use of old random_state during SPEC 7 transition
        rng = np.random.default_rng(724358723498249852)
        method = stats.BootstrapMethod(random_state=rng)
        res_ci = res.confidence_interval(method=method)
        ref_ci = res.confidence_interval()
        assert_allclose(res_ci, ref_ci, atol=1.5e-2)

        # `rng` is the new argument name`
        rng = np.random.default_rng(724358723498249852)
        method = stats.BootstrapMethod(rng=rng)
        res_ci2 = res.confidence_interval(method=method)
        assert_allclose(res_ci2, res_ci)

    @pytest.mark.parametrize('axis', [0, 1])
    def test_axis01(self, axis):
        rng = np.random.default_rng(38572345825)
        shape = (9, 10)
        x, y = rng.normal(size=(2,) + shape)
        res = stats.pearsonr(x, y, axis=axis)
        ci = res.confidence_interval()
        if axis == 0:
            x, y = x.T, y.T
        for i in range(x.shape[0]):
            res_i = stats.pearsonr(x[i], y[i])
            ci_i = res_i.confidence_interval()
            assert_allclose(res.statistic[i], res_i.statistic)
            assert_allclose(res.pvalue[i], res_i.pvalue)
            assert_allclose(ci.low[i], ci_i.low)
            assert_allclose(ci.high[i], ci_i.high)

    def test_axis_None(self):
        rng = np.random.default_rng(38572345825)
        shape = (9, 10)
        x, y = rng.normal(size=(2,) + shape)
        res = stats.pearsonr(x, y, axis=None)
        ci = res.confidence_interval()
        ref = stats.pearsonr(x.ravel(), y.ravel())
        ci_ref = ref.confidence_interval()
        assert_allclose(res.statistic, ref.statistic)
        assert_allclose(res.pvalue, ref.pvalue)
        assert_allclose(ci, ci_ref)

    def test_nd_input_validation(self, xp):
        x = y = xp.ones((2, 5))
        message = '`axis` must be an integer.'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, y, axis=1.5)

        message = '`x` and `y` must have the same length along `axis`'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, xp.ones((2, 1)), axis=1)

        message = '`x` and `y` must have length at least 2.'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(xp.ones((2, 1)), xp.ones((2, 1)), axis=1)

        message = '`x` and `y` must be broadcastable.'
        with pytest.raises(ValueError, match=message):
            stats.pearsonr(x, xp.ones((3, 5)), axis=1)

        message = '`method` must be `None` if arguments are not NumPy arrays.'
        if not is_numpy(xp):
            x = xp.arange(10)
            with pytest.raises(ValueError, match=message):
                stats.pearsonr(x, x, method=stats.PermutationMethod())

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_nd_special_cases(self, xp):
        rng = np.random.default_rng(34989235492245)

        x0, y0 = rng.random((4, 5)), rng.random((4, 5))
        x0[0, ...] = 1
        y0[1, ...] = 2
        x, y = xp.asarray(x0), xp.asarray(y0)
        message = 'An input array is constant'
        with eager_warns(stats.ConstantInputWarning, match=message, xp=xp):
            res = stats.pearsonr(x, y, axis=1)
            ci = res.confidence_interval()
            nans = xp.asarray([xp.nan, xp.nan], dtype=xp.float64)
            xp_assert_equal(res.statistic[0:2], nans)
            xp_assert_equal(res.pvalue[0:2], nans)
            xp_assert_equal(ci.low[0:2], nans)
            xp_assert_equal(ci.high[0:2], nans)
            assert xp.all(xp.isfinite(res.statistic[2:]))
            assert xp.all(xp.isfinite(res.pvalue[2:]))
            assert xp.all(xp.isfinite(ci.low[2:]))
            assert xp.all(xp.isfinite(ci.high[2:]))

        x0[0, 0], y0[1, 1] = 1 + 1e-15, 2 + 1e-15
        x, y = xp.asarray(x0), xp.asarray(y0)
        message = 'An input array is nearly constant'
        with eager_warns(stats.NearConstantInputWarning, match=message, xp=xp):
            stats.pearsonr(x, y, axis=1)

        # length 2 along axis
        x = xp.asarray([[1, 2], [1, 2], [2, 1], [2, 1.]])
        y = xp.asarray([[1, 2], [2, 1], [1, 2], [2, 1.]])
        ones = xp.ones(4)
        res = stats.pearsonr(x, y, axis=-1)
        ci = res.confidence_interval()
        xp_assert_close(res.statistic, xp.asarray([1, -1, -1, 1.]))
        xp_assert_close(res.pvalue, ones)
        xp_assert_close(ci.low, -ones)
        xp_assert_close(ci.high, ones)

    def test_different_dimensionality(self, xp):
        # For better or for worse, there is one difference between the broadcasting
        # behavior of most stats functions and NumPy gufuncs / NEP 5: gufuncs `axis`
        # refers to the core dimension *before* prepending `1`s to the array shapes
        # to match dimensionality; SciPy's prepends `1`s first. For instance, in
        # SciPy, `vecdot` would work just like `xp.sum(x * y, axis=axis)`, but this
        # is NOT true of NumPy. The discrepancy only arises when there are multiple
        # arguments with different dimensionality and positive indices are used,
        # which is probably why it hasn't been a problem. There are pros and cons of
        # each convention, and we might want to consider changing our behavior in
        # SciPy 2.0. For now, preserve consistency / backward compatibility.
        rng = np.random.default_rng(45834598265019344)
        x = rng.random((3, 10))
        y = rng.random(10)
        res = stats.pearsonr(x, y, axis=1)
        ref = stats.pearsonr(x, y, axis=-1)
        assert_equal(res.statistic, ref.statistic)

    @pytest.mark.parametrize('axis', [0, 1, None])
    @pytest.mark.parametrize('alternative', ['less', 'greater', 'two-sided'])
    def test_array_api(self, xp, axis, alternative):
        x, y = rng.normal(size=(2, 10, 11))
        res = stats.pearsonr(xp.asarray(x), xp.asarray(y),
                             axis=axis, alternative=alternative)
        ref = stats.pearsonr(x, y, axis=axis, alternative=alternative)
        xp_assert_close(res.statistic, xp.asarray(ref.statistic))
        xp_assert_close(res.pvalue, xp.asarray(ref.pvalue))

        res_ci = res.confidence_interval()
        ref_ci = ref.confidence_interval()
        xp_assert_close(res_ci.low, xp.asarray(ref_ci.low))
        xp_assert_close(res_ci.high, xp.asarray(ref_ci.high))


class TestFisherExact:
    """Some tests to show that fisher_exact() works correctly.

    Note that in SciPy 0.9.0 this was not working well for large numbers due to
    inaccuracy of the hypergeom distribution (see #1218). Fixed now.

    Also note that R and SciPy have different argument formats for their
    hypergeometric distribution functions.

    R:
    > phyper(18999, 99000, 110000, 39000, lower.tail = FALSE)
    [1] 1.701815e-09
    """

    def test_basic(self):
        fisher_exact = stats.fisher_exact

        res = fisher_exact([[14500, 20000], [30000, 40000]])[1]
        assert_approx_equal(res, 0.01106, significant=4)
        res = fisher_exact([[100, 2], [1000, 5]])[1]
        assert_approx_equal(res, 0.1301, significant=4)
        res = fisher_exact([[2, 7], [8, 2]])[1]
        assert_approx_equal(res, 0.0230141, significant=6)
        res = fisher_exact([[5, 1], [10, 10]])[1]
        assert_approx_equal(res, 0.1973244, significant=6)
        res = fisher_exact([[5, 15], [20, 20]])[1]
        assert_approx_equal(res, 0.0958044, significant=6)
        res = fisher_exact([[5, 16], [20, 25]])[1]
        assert_approx_equal(res, 0.1725862, significant=6)
        res = fisher_exact([[10, 5], [10, 1]])[1]
        assert_approx_equal(res, 0.1973244, significant=6)
        res = fisher_exact([[5, 0], [1, 4]])[1]
        assert_approx_equal(res, 0.04761904, significant=6)
        res = fisher_exact([[0, 1], [3, 2]])[1]
        assert_approx_equal(res, 1.0)
        res = fisher_exact([[0, 2], [6, 4]])[1]
        assert_approx_equal(res, 0.4545454545)
        res = fisher_exact([[2, 7], [8, 2]])
        assert_approx_equal(res[1], 0.0230141, significant=6)
        assert_approx_equal(res[0], 4.0 / 56)

    def test_precise(self):
        # results from R
        #
        # R defines oddsratio differently (see Notes section of fisher_exact
        # docstring), so those will not match.  We leave them in anyway, in
        # case they will be useful later on. We test only the p-value.
        tablist = [
            ([[100, 2], [1000, 5]], (2.505583993422285e-001, 1.300759363430016e-001)),
            ([[2, 7], [8, 2]], (8.586235135736206e-002, 2.301413756522114e-002)),
            ([[5, 1], [10, 10]], (4.725646047336584e+000, 1.973244147157190e-001)),
            ([[5, 15], [20, 20]], (3.394396617440852e-001, 9.580440012477637e-002)),
            ([[5, 16], [20, 25]], (3.960558326183334e-001, 1.725864953812994e-001)),
            ([[10, 5], [10, 1]], (2.116112781158483e-001, 1.973244147157190e-001)),
            ([[10, 5], [10, 0]], (0.000000000000000e+000, 6.126482213438734e-002)),
            ([[5, 0], [1, 4]], (np.inf, 4.761904761904762e-002)),
            ([[0, 5], [1, 4]], (0.000000000000000e+000, 1.000000000000000e+000)),
            ([[5, 1], [0, 4]], (np.inf, 4.761904761904758e-002)),
            ([[0, 1], [3, 2]], (0.000000000000000e+000, 1.000000000000000e+000))
            ]
        for table, res_r in tablist:
            res = stats.fisher_exact(np.asarray(table))
            np.testing.assert_almost_equal(res[1], res_r[1], decimal=11,
                                           verbose=True)

    def test_gh4130(self):
        # Previously, a fudge factor used to distinguish between theoretically
        # and numerically different probability masses was 1e-4; it has been
        # tightened to fix gh4130. Accuracy checked against R fisher.test.
        # options(digits=16)
        # table <- matrix(c(6, 108, 37, 200), nrow = 2)
        # fisher.test(table, alternative = "t")
        x = [[6, 37], [108, 200]]
        res = stats.fisher_exact(x)
        assert_allclose(res[1], 0.005092697748126)

        # case from https://github.com/brentp/fishers_exact_test/issues/27
        # That package has an (absolute?) fudge factor of 1e-6; too big
        x = [[22, 0], [0, 102]]
        res = stats.fisher_exact(x)
        assert_allclose(res[1], 7.175066786244549e-25)

        # case from https://github.com/brentp/fishers_exact_test/issues/1
        x = [[94, 48], [3577, 16988]]
        res = stats.fisher_exact(x)
        assert_allclose(res[1], 2.069356340993818e-37)

    def test_gh9231(self):
        # Previously, fisher_exact was extremely slow for this table
        # As reported in gh-9231, the p-value should be very nearly zero
        x = [[5829225, 5692693], [5760959, 5760959]]
        res = stats.fisher_exact(x)
        assert_allclose(res[1], 0, atol=1e-170)

    @pytest.mark.slow
    def test_large_numbers(self):
        # Test with some large numbers. Regression test for #1401
        pvals = [5.56e-11, 2.666e-11, 1.363e-11]  # from R
        for pval, num in zip(pvals, [75, 76, 77]):
            res = stats.fisher_exact([[17704, 496], [1065, num]])[1]
            assert_approx_equal(res, pval, significant=4)

        res = stats.fisher_exact([[18000, 80000], [20000, 90000]])[1]
        assert_approx_equal(res, 0.2751, significant=4)

    def test_raises(self):
        # test we raise an error for wrong number of dimensions.
        message = "The input `table` must have two dimensions."
        with pytest.raises(ValueError, match=message):
            stats.fisher_exact(np.arange(6))

    def test_row_or_col_zero(self):
        tables = ([[0, 0], [5, 10]],
                  [[5, 10], [0, 0]],
                  [[0, 5], [0, 10]],
                  [[5, 0], [10, 0]])
        for table in tables:
            oddsratio, pval = stats.fisher_exact(table)
            assert_equal(pval, 1.0)
            assert_equal(oddsratio, np.nan)

    def test_less_greater(self):
        tables = (
            # Some tables to compare with R:
            [[2, 7], [8, 2]],
            [[200, 7], [8, 300]],
            [[28, 21], [6, 1957]],
            [[190, 800], [200, 900]],
            # Some tables with simple exact values
            # (includes regression test for ticket #1568):
            [[0, 2], [3, 0]],
            [[1, 1], [2, 1]],
            [[2, 0], [1, 2]],
            [[0, 1], [2, 3]],
            [[1, 0], [1, 4]],
            )
        pvals = (
            # from R:
            [0.018521725952066501, 0.9990149169715733],
            [1.0, 2.0056578803889148e-122],
            [1.0, 5.7284374608319831e-44],
            [0.7416227, 0.2959826],
            # Exact:
            [0.1, 1.0],
            [0.7, 0.9],
            [1.0, 0.3],
            [2./3, 1.0],
            [1.0, 1./3],
            )
        for table, pval in zip(tables, pvals):
            res = []
            res.append(stats.fisher_exact(table, alternative="less")[1])
            res.append(stats.fisher_exact(table, alternative="greater")[1])
            assert_allclose(res, pval, atol=0, rtol=1e-7)

    def test_gh3014(self):
        # check if issue #3014 has been fixed.
        # before, this would have risen a ValueError
        odds, pvalue = stats.fisher_exact([[1, 2], [9, 84419233]])

    @pytest.mark.parametrize("alternative", ['two-sided', 'less', 'greater'])
    def test_result(self, alternative):
        table = np.array([[14500, 20000], [30000, 40000]])
        res = stats.fisher_exact(table, alternative=alternative)
        assert_equal((res.statistic, res.pvalue), res)

    def test_input_validation_edge_cases_rxc(self):
        rng = np.random.default_rng(2345783457834572345)
        table = np.asarray([[2, 7], [8, 2]])

        message = r"`alternative` must be the default \(None\) unless..."
        with pytest.raises(ValueError, match=message):
            method = stats.PermutationMethod(rng=rng)
            stats.fisher_exact(table, method=method, alternative='less')

        message = "...not recognized; if provided, `method` must be an..."
        with pytest.raises(ValueError, match=message):
            method = stats.BootstrapMethod(rng=rng)
            stats.fisher_exact(table, method=method)

        message = "If the `method` argument of `fisher_exact` is an..."
        with pytest.raises(ValueError, match=message):
            method = stats.MonteCarloMethod(rvs=stats.norm.rvs)
            stats.fisher_exact(table, method=method)

        message = "`table` must have at least one row and one column."
        with pytest.raises(ValueError, match=message):
            stats.fisher_exact(np.zeros((0, 1)))

        # Specical case: when there is only one table with given marginals, the
        # PMF of that case is 1.0, so the p-value is 1.0
        np.testing.assert_equal(stats.fisher_exact([[1, 2, 3]]), (1, 1))
        np.testing.assert_equal(stats.fisher_exact([[1], [2], [3]]), (1, 1))
        np.testing.assert_equal(stats.fisher_exact(np.zeros((2, 3))), (1, 1))



    @pytest.mark.fail_slow(10)
    @pytest.mark.slow()
    def test_resampling_2x2(self):
        rng = np.random.default_rng(2345783457834572345)
        table = np.asarray([[2, 7], [8, 2]])
        ref = stats.fisher_exact(table)
        ref_pvalue = ref.pvalue
        ref_stat = stats.random_table(table.sum(axis=1), table.sum(axis=0)).pmf(table)

        method = stats.MonteCarloMethod(rng=rng)
        res = stats.fisher_exact(table, method=method)
        assert_allclose(res.pvalue, ref_pvalue, atol=0.0025)
        assert_equal(res.statistic, ref_stat)

        method = stats.PermutationMethod(rng=rng)
        res = stats.fisher_exact(table, method=method)
        assert_allclose(res.pvalue, ref.pvalue, atol=0.0025)
        assert_equal(res.statistic, ref_stat)

    @pytest.mark.fail_slow(10)
    @pytest.mark.slow()
    def test_resampling_rxc(self):
        # Compare against R fisher.exact
        # options(digits=16)
        # MP6 < - rbind(
        #     c(1, 2, 2, 1, 1, 0, 1),
        #     c(2, 0, 0, 2, 3, 0, 0),
        #     c(0, 1, 1, 1, 2, 7, 3),
        #     c(1, 1, 2, 0, 0, 0, 1),
        #     c(0, 1, 1, 1, 1, 0, 0))
        # fisher.test(MP6)

        table = [[1, 2, 2, 1, 1, 0, 1],
                 [2, 0, 0, 2, 3, 0, 0],
                 [0, 1, 1, 1, 2, 7, 3],
                 [1, 1, 2, 0, 0, 0, 1],
                 [0, 1, 1, 1, 1, 0, 0]]
        table = np.asarray(table)

        ref_pvalue = 0.03928964365533
        rng = np.random.default_rng(3928964365533)

        method = stats.PermutationMethod(rng=rng)
        res = stats.fisher_exact(table, method=method)
        assert_allclose(res.pvalue, ref_pvalue, atol=5e-4)

        method = stats.MonteCarloMethod(rng=rng, n_resamples=99999)
        res = stats.fisher_exact(table, method=method)
        assert_allclose(res.pvalue, ref_pvalue, atol=5e-4)

    @pytest.mark.xslow()
    def test_resampling_exact_2x2(self):
        # Test that exact permutation p-value matches result of `fisher_exact`
        rng = np.random.default_rng(2345783457834572345)
        method = stats.PermutationMethod(rng=rng)

        for a in range(1, 3):
            for b in range(1, 3):
                for c in range(1, 3):
                    for d in range(1, 4):
                        table = np.asarray([[a, b], [c, d]])
                        ref = stats.fisher_exact(table)
                        res = stats.fisher_exact(table, method=method)
                        assert_allclose(res.pvalue, ref.pvalue, atol=1e-14)


class TestCorrSpearmanr:
    """ W.II.D. Compute a correlation matrix on all the variables.

        All the correlations, except for ZERO and MISS, should be exactly 1.
        ZERO and MISS should have undefined or missing correlations with the
        other variables.  The same should go for SPEARMAN correlations, if
        your program has them.
    """

    def setup_method(self):
        self.rng = np.random.default_rng(228584263)

    def test_scalar(self):
        y = stats.spearmanr(4., 2.)
        assert_(np.isnan(y).all())

    def test_uneven_lengths(self):
        assert_raises(ValueError, stats.spearmanr, [1, 2, 1], [8, 9])
        assert_raises(ValueError, stats.spearmanr, [1, 2, 1], 8)

    def test_uneven_2d_shapes(self):
        # Different number of columns should work - those just get concatenated.
        x = self.rng.standard_normal((4, 3))
        y = self.rng.standard_normal((4, 2))
        assert stats.spearmanr(x, y).statistic.shape == (5, 5)
        assert stats.spearmanr(x.T, y.T, axis=1).pvalue.shape == (5, 5)

        assert_raises(ValueError, stats.spearmanr, x, y, axis=1)
        assert_raises(ValueError, stats.spearmanr, x.T, y.T)

    def test_ndim_too_high(self):
        x = self.rng.standard_normal((4, 3, 2))
        assert_raises(ValueError, stats.spearmanr, x)
        assert_raises(ValueError, stats.spearmanr, x, x)
        assert_raises(ValueError, stats.spearmanr, x, None, None)
        # But should work with axis=None (raveling axes) for two input arrays
        assert_allclose(stats.spearmanr(x, x, axis=None),
                        stats.spearmanr(x.flatten(), x.flatten(), axis=0))

    def test_nan_policy(self):
        x = np.arange(10.)
        x[9] = np.nan
        assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
        assert_array_equal(stats.spearmanr(x, x, nan_policy='omit'),
                           (1.0, 0.0))
        assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
        assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')

    def test_nan_policy_bug_12458(self):
        rng = np.random.default_rng(8119864466)
        x = rng.random((5, 10))
        k = 6
        x[:, k] = np.nan
        y = np.delete(x, k, axis=1)
        corx, px = stats.spearmanr(x, nan_policy='omit')
        cory, py = stats.spearmanr(y)
        corx = np.delete(np.delete(corx, k, axis=1), k, axis=0)
        px = np.delete(np.delete(px, k, axis=1), k, axis=0)
        assert_allclose(corx, cory, atol=1e-14)
        assert_allclose(px, py, atol=1e-14)

    def test_nan_policy_bug_12411(self):
        m = 5
        n = 10
        x = self.rng.standard_normal((m, n))
        x[1, 0] = np.nan
        x[3, -1] = np.nan
        corr, pvalue = stats.spearmanr(x, axis=1, nan_policy="propagate")
        res = [[stats.spearmanr(x[i, :], x[j, :]).statistic for i in range(m)]
               for j in range(m)]
        assert_allclose(corr, res)

    def test_sXX(self):
        y = stats.spearmanr(X,X)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXBIG(self):
        y = stats.spearmanr(X,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXLITTLE(self):
        y = stats.spearmanr(X,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXHUGE(self):
        y = stats.spearmanr(X,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXTINY(self):
        y = stats.spearmanr(X,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sXROUND(self):
        y = stats.spearmanr(X,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGBIG(self):
        y = stats.spearmanr(BIG,BIG)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGLITTLE(self):
        y = stats.spearmanr(BIG,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGHUGE(self):
        y = stats.spearmanr(BIG,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGTINY(self):
        y = stats.spearmanr(BIG,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sBIGROUND(self):
        y = stats.spearmanr(BIG,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLELITTLE(self):
        y = stats.spearmanr(LITTLE,LITTLE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLEHUGE(self):
        y = stats.spearmanr(LITTLE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLETINY(self):
        y = stats.spearmanr(LITTLE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sLITTLEROUND(self):
        y = stats.spearmanr(LITTLE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sHUGEHUGE(self):
        y = stats.spearmanr(HUGE,HUGE)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sHUGETINY(self):
        y = stats.spearmanr(HUGE,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sHUGEROUND(self):
        y = stats.spearmanr(HUGE,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sTINYTINY(self):
        y = stats.spearmanr(TINY,TINY)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sTINYROUND(self):
        y = stats.spearmanr(TINY,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_sROUNDROUND(self):
        y = stats.spearmanr(ROUND,ROUND)
        r = y[0]
        assert_approx_equal(r,1.0)

    def test_spearmanr_result_attributes(self):
        res = stats.spearmanr(X, X)
        attributes = ('correlation', 'pvalue')
        check_named_results(res, attributes)
        assert_equal(res.correlation, res.statistic)

    def test_1d_vs_2d(self):
        x1 = [1, 2, 3, 4, 5, 6]
        x2 = [1, 2, 3, 4, 6, 5]
        res1 = stats.spearmanr(x1, x2)
        res2 = stats.spearmanr(np.asarray([x1, x2]).T)
        assert_allclose(res1, res2)

    def test_1d_vs_2d_nans(self):
        # Now the same with NaNs present.  Regression test for gh-9103.
        for nan_policy in ['propagate', 'omit']:
            x1 = [1, np.nan, 3, 4, 5, 6]
            x2 = [1, 2, 3, 4, 6, np.nan]
            res1 = stats.spearmanr(x1, x2, nan_policy=nan_policy)
            res2 = stats.spearmanr(np.asarray([x1, x2]).T, nan_policy=nan_policy)
            assert_allclose(res1, res2)

    def test_3cols(self):
        x1 = np.arange(6)
        x2 = -x1
        x3 = np.array([0, 1, 2, 3, 5, 4])
        x = np.asarray([x1, x2, x3]).T
        actual = stats.spearmanr(x)
        expected_corr = np.array([[1, -1, 0.94285714],
                                  [-1, 1, -0.94285714],
                                  [0.94285714, -0.94285714, 1]])
        expected_pvalue = np.zeros((3, 3), dtype=float)
        expected_pvalue[2, 0:2] = 0.00480466472
        expected_pvalue[0:2, 2] = 0.00480466472

        assert_allclose(actual.statistic, expected_corr)
        assert_allclose(actual.pvalue, expected_pvalue)

    def test_gh_9103(self):
        # Regression test for gh-9103.
        x = np.array([[np.nan, 3.0, 4.0, 5.0, 5.1, 6.0, 9.2],
                      [5.0, np.nan, 4.1, 4.8, 4.9, 5.0, 4.1],
                      [0.5, 4.0, 7.1, 3.8, 8.0, 5.1, 7.6]]).T
        corr = np.array([[np.nan, np.nan, np.nan],
                         [np.nan, np.nan, np.nan],
                         [np.nan, np.nan, 1.]])
        assert_allclose(stats.spearmanr(x, nan_policy='propagate').statistic,
                        corr)

        res = stats.spearmanr(x, nan_policy='omit').statistic
        assert_allclose((res[0][1], res[0][2], res[1][2]),
                        (0.2051957, 0.4857143, -0.4707919), rtol=1e-6)

    def test_gh_8111(self):
        # Regression test for gh-8111 (different result for float/int/bool).
        n = 100
        rng = np.random.RandomState(234568)
        x = rng.rand(n)
        m = rng.rand(n) > 0.7

        # bool against float, no nans
        a = (x > .5)
        b = np.array(x)
        res1 = stats.spearmanr(a, b, nan_policy='omit').statistic

        # bool against float with NaNs
        b[m] = np.nan
        res2 = stats.spearmanr(a, b, nan_policy='omit').statistic

        # int against float with NaNs
        a = a.astype(np.int32)
        res3 = stats.spearmanr(a, b, nan_policy='omit').statistic

        expected = [0.865895477, 0.866100381, 0.866100381]
        assert_allclose([res1, res2, res3], expected)


class TestCorrSpearmanr2:
    """Some further tests of the spearmanr function."""

    def test_spearmanr_vs_r(self):
        # Cross-check with R:
        # cor.test(c(1,2,3,4,5),c(5,6,7,8,7),method="spearmanr")
        x1 = [1, 2, 3, 4, 5]
        x2 = [5, 6, 7, 8, 7]
        expected = (0.82078268166812329, 0.088587005313543798)
        res = stats.spearmanr(x1, x2)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    def test_empty_arrays(self):
        assert_equal(stats.spearmanr([], []), (np.nan, np.nan))

    def test_normal_draws(self):
        rng = np.random.RandomState(7546)
        x = np.array([rng.normal(loc=1, scale=1, size=500),
                      rng.normal(loc=1, scale=1, size=500)])
        corr = [[1.0, 0.3],
                [0.3, 1.0]]
        x = np.dot(np.linalg.cholesky(corr), x)
        expected = (0.28659685838743354, 6.579862219051161e-11)
        res = stats.spearmanr(x[0], x[1])
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    def test_corr_1(self):
        assert_approx_equal(stats.spearmanr([1, 1, 2], [1, 1, 2])[0], 1.0)

    def test_nan_policies(self):
        x = np.arange(10.)
        x[9] = np.nan
        assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
        assert_allclose(stats.spearmanr(x, x, nan_policy='omit'),
                        (1.0, 0))
        assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
        assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')

    def test_unequal_lengths(self):
        x = np.arange(10.)
        y = np.arange(20.)
        assert_raises(ValueError, stats.spearmanr, x, y)

    def test_omit_paired_value(self):
        x1 = [1, 2, 3, 4]
        x2 = [8, 7, 6, np.nan]
        res1 = stats.spearmanr(x1, x2, nan_policy='omit')
        res2 = stats.spearmanr(x1[:3], x2[:3], nan_policy='omit')
        assert_equal(res1, res2)

    def test_gh_issue_6061_windows_overflow(self):
        x = list(range(2000))
        y = list(range(2000))
        y[0], y[9] = y[9], y[0]
        y[10], y[434] = y[434], y[10]
        y[435], y[1509] = y[1509], y[435]
        # rho = 1 - 6 * (2 * (9^2 + 424^2 + 1074^2))/(2000 * (2000^2 - 1))
        #     = 1 - (1 / 500)
        #     = 0.998
        x.append(np.nan)
        y.append(3.0)
        assert_almost_equal(stats.spearmanr(x, y, nan_policy='omit')[0], 0.998)

    def test_tie0(self):
        # with only ties in one or both inputs
        warn_msg = "An input array is constant"
        with pytest.warns(stats.ConstantInputWarning, match=warn_msg):
            r, p = stats.spearmanr([2, 2, 2], [2, 2, 2])
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)
            r, p = stats.spearmanr([2, 0, 2], [2, 2, 2])
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)
            r, p = stats.spearmanr([2, 2, 2], [2, 0, 2])
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)

    def test_tie1(self):
        # Data
        x = [1.0, 2.0, 3.0, 4.0]
        y = [1.0, 2.0, 2.0, 3.0]
        # Ranks of the data, with tie-handling.
        xr = [1.0, 2.0, 3.0, 4.0]
        yr = [1.0, 2.5, 2.5, 4.0]
        # Result of spearmanr should be the same as applying
        # pearsonr to the ranks.
        sr = stats.spearmanr(x, y)
        pr = stats.pearsonr(xr, yr)
        assert_almost_equal(sr, pr)

    def test_tie2(self):
        # Test tie-handling if inputs contain nan's
        # Data without nan's
        x1 = [1, 2, 2.5, 2]
        y1 = [1, 3, 2.5, 4]
        # Same data with nan's
        x2 = [1, 2, 2.5, 2, np.nan]
        y2 = [1, 3, 2.5, 4, np.nan]

        # Results for two data sets should be the same if nan's are ignored
        sr1 = stats.spearmanr(x1, y1)
        sr2 = stats.spearmanr(x2, y2, nan_policy='omit')
        assert_almost_equal(sr1, sr2)

    def test_ties_axis_1(self):
        z1 = np.array([[1, 1, 1, 1], [1, 2, 3, 4]])
        z2 = np.array([[1, 2, 3, 4], [1, 1, 1, 1]])
        z3 = np.array([[1, 1, 1, 1], [1, 1, 1, 1]])
        warn_msg = "An input array is constant"
        with pytest.warns(stats.ConstantInputWarning, match=warn_msg):
            r, p = stats.spearmanr(z1, axis=1)
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)
            r, p = stats.spearmanr(z2, axis=1)
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)
            r, p = stats.spearmanr(z3, axis=1)
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)

    def test_gh_11111(self):
        x = np.array([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
        y = np.array([0, 0.009783728115345005, 0, 0, 0.0019759230121848587,
                      0.0007535430349118562, 0.0002661781514710257, 0, 0,
                      0.0007835762419683435])
        warn_msg = "An input array is constant"
        with pytest.warns(stats.ConstantInputWarning, match=warn_msg):
            r, p = stats.spearmanr(x, y)
            assert_equal(r, np.nan)
            assert_equal(p, np.nan)

    def test_index_error(self):
        x = np.array([1.0, 7.0, 2.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
        y = np.array([0, 0.009783728115345005, 0, 0, 0.0019759230121848587,
                      0.0007535430349118562, 0.0002661781514710257, 0, 0,
                      0.0007835762419683435])
        assert_raises(ValueError, stats.spearmanr, x, y, axis=2)

    def test_alternative(self):
        # Test alternative parameter

        # Simple test - Based on the above ``test_spearmanr_vs_r``
        x1 = [1, 2, 3, 4, 5]
        x2 = [5, 6, 7, 8, 7]

        # strong positive correlation
        expected = (0.82078268166812329, 0.088587005313543798)

        # correlation > 0 -> large "less" p-value
        res = stats.spearmanr(x1, x2, alternative="less")
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], 1 - (expected[1] / 2))

        # correlation > 0 -> small "less" p-value
        res = stats.spearmanr(x1, x2, alternative="greater")
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1] / 2)

        with pytest.raises(ValueError, match="`alternative` must be 'less'..."):
            stats.spearmanr(x1, x2, alternative="ekki-ekki")

    @pytest.mark.parametrize("alternative", ('two-sided', 'less', 'greater'))
    def test_alternative_nan_policy(self, alternative):
        # Test nan policies
        x1 = [1, 2, 3, 4, 5]
        x2 = [5, 6, 7, 8, 7]
        x1nan = x1 + [np.nan]
        x2nan = x2 + [np.nan]

        # test nan_policy="propagate"
        assert_array_equal(stats.spearmanr(x1nan, x2nan), (np.nan, np.nan))

        # test nan_policy="omit"
        res_actual = stats.spearmanr(x1nan, x2nan, nan_policy='omit',
                                     alternative=alternative)
        res_expected = stats.spearmanr(x1, x2, alternative=alternative)
        assert_allclose(res_actual, res_expected)

        # test nan_policy="raise"
        message = 'The input contains nan values'
        with pytest.raises(ValueError, match=message):
            stats.spearmanr(x1nan, x2nan, nan_policy='raise',
                            alternative=alternative)

        # test invalid nan_policy
        message = "nan_policy must be one of..."
        with pytest.raises(ValueError, match=message):
            stats.spearmanr(x1nan, x2nan, nan_policy='ekki-ekki',
                            alternative=alternative)


#    W.II.E.  Tabulate X against X, using BIG as a case weight.  The values
#    should appear on the diagonal and the total should be 899999955.
#    If the table cannot hold these values, forget about working with
#    census data.  You can also tabulate HUGE against TINY.  There is no
#    reason a tabulation program should not be able to distinguish
#    different values regardless of their magnitude.

# I need to figure out how to do this one.

@pytest.mark.thread_unsafe(reason="fails in parallel")
def test_kendalltau():
    # For the cases without ties, both variants should give the same
    # result.
    variants = ('b', 'c')

    # case without ties, con-dis equal zero
    x = [5, 2, 1, 3, 6, 4, 7, 8]
    y = [5, 2, 6, 3, 1, 8, 7, 4]
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.0, 1.0)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # case without ties, con-dis equal zero
    x = [0, 5, 2, 1, 3, 6, 4, 7, 8]
    y = [5, 2, 0, 6, 3, 1, 8, 7, 4]
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.0, 1.0)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # case without ties, con-dis close to zero
    x = [5, 2, 1, 3, 6, 4, 7]
    y = [5, 2, 6, 3, 1, 7, 4]
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-0.14285714286, 0.77261904762)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # case without ties, con-dis close to zero
    x = [2, 1, 3, 6, 4, 7, 8]
    y = [2, 6, 3, 1, 8, 7, 4]
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.047619047619, 1.0)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # simple case without ties
    x = np.arange(10)
    y = np.arange(10)
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (1.0, 5.511463844797e-07)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # swap a couple of values
    b = y[1]
    y[1] = y[2]
    y[2] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.9555555555555556, 5.511463844797e-06)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # swap a couple more
    b = y[5]
    y[5] = y[6]
    y[6] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (0.9111111111111111, 2.976190476190e-05)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # same in opposite direction
    x = np.arange(10)
    y = np.arange(10)[::-1]
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-1.0, 5.511463844797e-07)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # swap a couple of values
    b = y[1]
    y[1] = y[2]
    y[2] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-0.9555555555555556, 5.511463844797e-06)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # swap a couple more
    b = y[5]
    y[5] = y[6]
    y[6] = b
    # Cross-check with exact result from R:
    # cor.test(x,y,method="kendall",exact=1)
    expected = (-0.9111111111111111, 2.976190476190e-05)
    for taux in variants:
        res = stats.kendalltau(x, y, variant=taux)
        assert_approx_equal(res[0], expected[0])
        assert_approx_equal(res[1], expected[1])

    # Check a case where variants are different
    # Example values found from Kendall (1970).
    # P-value is the same for the both variants
    x = array([1, 2, 2, 4, 4, 6, 6, 8, 9, 9])
    y = array([1, 2, 4, 4, 4, 4, 8, 8, 8, 10])
    expected = 0.85895569
    assert_approx_equal(stats.kendalltau(x, y, variant='b')[0], expected)
    expected = 0.825
    assert_approx_equal(stats.kendalltau(x, y, variant='c')[0], expected)

    # check exception in case of ties and method='exact' requested
    y[2] = y[1]
    assert_raises(ValueError, stats.kendalltau, x, y, method='exact')

    # check exception in case of invalid method keyword
    assert_raises(ValueError, stats.kendalltau, x, y, method='banana')

    # check exception in case of invalid variant keyword
    assert_raises(ValueError, stats.kendalltau, x, y, variant='rms')

    # tau-b with some ties
    # Cross-check with R:
    # cor.test(c(12,2,1,12,2),c(1,4,7,1,0),method="kendall",exact=FALSE)
    x1 = [12, 2, 1, 12, 2]
    x2 = [1, 4, 7, 1, 0]
    expected = (-0.47140452079103173, 0.28274545993277478)
    res = stats.kendalltau(x1, x2)
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # test for namedtuple attribute results
    attributes = ('correlation', 'pvalue')
    for taux in variants:
        res = stats.kendalltau(x1, x2, variant=taux)
        check_named_results(res, attributes)
        assert_equal(res.correlation, res.statistic)

    # with only ties in one or both inputs in tau-b or tau-c
    for taux in variants:
        assert_equal(stats.kendalltau([2, 2, 2], [2, 2, 2], variant=taux),
                     (np.nan, np.nan))
        assert_equal(stats.kendalltau([2, 0, 2], [2, 2, 2], variant=taux),
                     (np.nan, np.nan))
        assert_equal(stats.kendalltau([2, 2, 2], [2, 0, 2], variant=taux),
                     (np.nan, np.nan))

    # empty arrays provided as input
    with pytest.warns(SmallSampleWarning, match="One or more sample..."):
        assert_equal(stats.kendalltau([], []), (np.nan, np.nan))

    # check with larger arrays
    rng = np.random.RandomState(7546)
    x = np.array([rng.normal(loc=1, scale=1, size=500),
                  rng.normal(loc=1, scale=1, size=500)])
    corr = [[1.0, 0.3],
            [0.3, 1.0]]
    x = np.dot(np.linalg.cholesky(corr), x)
    expected = (0.19291382765531062, 1.1337095377742629e-10)
    res = stats.kendalltau(x[0], x[1])
    assert_approx_equal(res[0], expected[0])
    assert_approx_equal(res[1], expected[1])

    # this should result in 1 for taub but not tau-c
    assert_approx_equal(stats.kendalltau([1, 1, 2], [1, 1, 2], variant='b')[0],
                        1.0)
    assert_approx_equal(stats.kendalltau([1, 1, 2], [1, 1, 2], variant='c')[0],
                        0.88888888)

    # test nan_policy
    x = np.arange(10.)
    x[9] = np.nan
    assert_array_equal(stats.kendalltau(x, x), (np.nan, np.nan))
    assert_allclose(stats.kendalltau(x, x, nan_policy='omit'),
                    (1.0, 5.5114638e-6), rtol=1e-06)
    assert_allclose(stats.kendalltau(x, x, nan_policy='omit', method='asymptotic'),
                    (1.0, 0.00017455009626808976), rtol=1e-06)
    assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='raise')
    assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='foobar')

    # test unequal length inputs
    x = np.arange(10.)
    y = np.arange(20.)
    assert_raises(ValueError, stats.kendalltau, x, y)

    # test all ties
    with pytest.warns(SmallSampleWarning, match="One or more sample..."):
        tau, p_value = stats.kendalltau([0], [0])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)

    # Regression test for GitHub issue #6061 - Overflow on Windows
    x = np.arange(2000, dtype=float)
    x = np.ma.masked_greater(x, 1995)
    y = np.arange(2000, dtype=float)
    y = np.concatenate((y[1000:], y[:1000]))
    assert_(np.isfinite(stats.mstats.kendalltau(x,y)[1]))


def test_kendalltau_vs_mstats_basic():
    rng = np.random.RandomState(42)
    for s in range(3, 10):
        a = []
        # Generate rankings with ties
        for i in range(s):
            a += [i]*i
        b = list(a)
        rng.shuffle(a)
        rng.shuffle(b)
        expected = mstats_basic.kendalltau(a, b)
        actual = stats.kendalltau(a, b)
        assert_approx_equal(actual[0], expected[0])
        assert_approx_equal(actual[1], expected[1])


def test_kendalltau_nan_2nd_arg():
    # regression test for gh-6134: nans in the second arg were not handled
    x = [1., 2., 3., 4.]
    y = [np.nan, 2.4, 3.4, 3.4]

    r1 = stats.kendalltau(x, y, nan_policy='omit')
    r2 = stats.kendalltau(x[1:], y[1:])
    assert_allclose(r1.statistic, r2.statistic, atol=1e-15)


@pytest.mark.thread_unsafe(reason="fails in parallel")
def test_kendalltau_gh18139_overflow():
    # gh-18139 reported an overflow in `kendalltau` that appeared after
    # SciPy 0.15.1. Check that this particular overflow does not occur.
    # (Test would fail if warning were emitted.)
    import random
    random.seed(6272161)
    classes = [1, 2, 3, 4, 5, 6, 7]
    n_samples = 2 * 10 ** 5
    x = random.choices(classes, k=n_samples)
    y = random.choices(classes, k=n_samples)
    res = stats.kendalltau(x, y)
    # Reference value from SciPy 0.15.1
    assert_allclose(res.statistic, 0.0011816493905730343)
    # Reference p-value from `permutation_test` w/ n_resamples=9999 (default).
    # Expected to be accurate to at least two digits.
    assert_allclose(res.pvalue, 0.4894, atol=2e-3)


class TestKendallTauAlternative:
    def test_kendalltau_alternative_asymptotic(self):
        # Test alternative parameter, asymptotic method (due to tie)

        # Based on TestCorrSpearman2::test_alternative
        x1 = [1, 2, 3, 4, 5]
        x2 = [5, 6, 7, 8, 7]

        # strong positive correlation
        expected = stats.kendalltau(x1, x2, alternative="two-sided")
        assert expected[0] > 0

        # rank correlation > 0 -> large "less" p-value
        res = stats.kendalltau(x1, x2, alternative="less")
        assert_equal(res[0], expected[0])
        assert_allclose(res[1], 1 - (expected[1] / 2))

        # rank correlation > 0 -> small "greater" p-value
        res = stats.kendalltau(x1, x2, alternative="greater")
        assert_equal(res[0], expected[0])
        assert_allclose(res[1], expected[1] / 2)

        # reverse the direction of rank correlation
        x2.reverse()

        # strong negative correlation
        expected = stats.kendalltau(x1, x2, alternative="two-sided")
        assert expected[0] < 0

        # rank correlation < 0 -> large "greater" p-value
        res = stats.kendalltau(x1, x2, alternative="greater")
        assert_equal(res[0], expected[0])
        assert_allclose(res[1], 1 - (expected[1] / 2))

        # rank correlation < 0 -> small "less" p-value
        res = stats.kendalltau(x1, x2, alternative="less")
        assert_equal(res[0], expected[0])
        assert_allclose(res[1], expected[1] / 2)

        with pytest.raises(ValueError, match="`alternative` must be 'less'..."):
            stats.kendalltau(x1, x2, alternative="ekki-ekki")

    # There are a lot of special cases considered in the calculation of the
    # exact p-value, so we test each separately. We also need to test
    # separately when the observed statistic is in the left tail vs the right
    # tail because the code leverages symmetry of the null distribution; to
    # do that we use the same test case but negate one of the samples.
    # Reference values computed using R cor.test, e.g.
    # options(digits=16)
    # x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
    # y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
    # cor.test(x, y, method = "kendall", alternative = "g")

    alternatives = ('less', 'two-sided', 'greater')
    p_n1 = [np.nan, np.nan, np.nan]
    p_n2 = [1, 1, 0.5]
    p_c0 = [1, 0.3333333333333, 0.1666666666667]
    p_c1 = [0.9583333333333, 0.3333333333333, 0.1666666666667]
    p_no_correlation = [0.5916666666667, 1, 0.5916666666667]
    p_no_correlationb = [0.5475694444444, 1, 0.5475694444444]
    p_n_lt_171 = [0.9624118165785, 0.1194389329806, 0.0597194664903]
    p_n_lt_171b = [0.246236925303, 0.4924738506059, 0.755634083327]
    p_n_lt_171c = [0.9847475308925, 0.03071385306533, 0.01535692653267]

    def exact_test(self, x, y, alternative, rev, stat_expected, p_expected):
        if rev:
            y = -np.asarray(y)
            stat_expected *= -1
        res = stats.kendalltau(x, y, method='exact', alternative=alternative)
        res_expected = stat_expected, p_expected
        assert_allclose(res, res_expected)

    case_R_n1 = (list(zip(alternatives, p_n1, [False]*3))
                 + list(zip(alternatives, reversed(p_n1), [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_n1)
    def test_against_R_n1(self, alternative, p_expected, rev):
        x, y = [1], [2]
        stat_expected = np.nan
        with pytest.warns(SmallSampleWarning, match="One or more sample..."):
            self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_n2 = (list(zip(alternatives, p_n2, [False]*3))
                 + list(zip(alternatives, reversed(p_n2), [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_n2)
    def test_against_R_n2(self, alternative, p_expected, rev):
        x, y = [1, 2], [3, 4]
        stat_expected = 0.9999999999999998
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_c0 = (list(zip(alternatives, p_c0, [False]*3))
                 + list(zip(alternatives, reversed(p_c0), [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_c0)
    def test_against_R_c0(self, alternative, p_expected, rev):
        x, y = [1, 2, 3], [1, 2, 3]
        stat_expected = 1
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_c1 = (list(zip(alternatives, p_c1, [False]*3))
                 + list(zip(alternatives, reversed(p_c1), [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_c1)
    def test_against_R_c1(self, alternative, p_expected, rev):
        x, y = [1, 2, 3, 4], [1, 2, 4, 3]
        stat_expected = 0.6666666666666667
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_no_corr = (list(zip(alternatives, p_no_correlation, [False]*3))
                      + list(zip(alternatives, reversed(p_no_correlation),
                                 [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_no_corr)
    def test_against_R_no_correlation(self, alternative, p_expected, rev):
        x, y = [1, 2, 3, 4, 5], [1, 5, 4, 2, 3]
        stat_expected = 0
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_no_cor_b = (list(zip(alternatives, p_no_correlationb, [False]*3))
                     + list(zip(alternatives, reversed(p_no_correlationb),
                                [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_no_cor_b)
    def test_against_R_no_correlationb(self, alternative, p_expected, rev):
        x, y = [1, 2, 3, 4, 5, 6, 7, 8], [8, 6, 1, 3, 2, 5, 4, 7]
        stat_expected = 0
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_lt_171 = (list(zip(alternatives, p_n_lt_171, [False]*3))
                     + list(zip(alternatives, reversed(p_n_lt_171), [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_lt_171)
    def test_against_R_lt_171(self, alternative, p_expected, rev):
        # Data from Hollander & Wolfe (1973), p. 187f.
        # Used from https://rdrr.io/r/stats/cor.test.html
        x = [44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1]
        y = [2.6, 3.1, 2.5, 5.0, 3.6, 4.0, 5.2, 2.8, 3.8]
        stat_expected = 0.4444444444444445
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_lt_171b = (list(zip(alternatives, p_n_lt_171b, [False]*3))
                      + list(zip(alternatives, reversed(p_n_lt_171b),
                                 [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_lt_171b)
    def test_against_R_lt_171b(self, alternative, p_expected, rev):
        rng = np.random.RandomState(0)
        x = rng.rand(100)
        y = rng.rand(100)
        stat_expected = -0.04686868686868687
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_R_lt_171c = (list(zip(alternatives, p_n_lt_171c, [False]*3))
                      + list(zip(alternatives, reversed(p_n_lt_171c),
                                 [True]*3)))

    @pytest.mark.parametrize("alternative, p_expected, rev", case_R_lt_171c)
    def test_against_R_lt_171c(self, alternative, p_expected, rev):
        rng = np.random.RandomState(0)
        x = rng.rand(170)
        y = rng.rand(170)
        stat_expected = 0.1115906717716673
        self.exact_test(x, y, alternative, rev, stat_expected, p_expected)

    case_gt_171 = (list(zip(alternatives, [False]*3)) +
                   list(zip(alternatives, [True]*3)))

    @pytest.mark.parametrize("alternative, rev", case_gt_171)
    def test_gt_171(self, alternative, rev):
        rng = np.random.RandomState(0)
        x = rng.rand(400)
        y = rng.rand(400)
        res0 = stats.kendalltau(x, y, method='exact',
                                alternative=alternative)
        res1 = stats.kendalltau(x, y, method='asymptotic',
                                alternative=alternative)
        assert_equal(res0[0], res1[0])
        assert_allclose(res0[1], res1[1], rtol=1e-3)

    @pytest.mark.parametrize("method", ('exact', 'asymptotic'))
    @pytest.mark.parametrize("alternative", ('two-sided', 'less', 'greater'))
    def test_nan_policy(self, method, alternative):
        # Test nan policies
        x1 = [1, 2, 3, 4, 5]
        x2 = [5, 6, 7, 8, 9]
        x1nan = x1 + [np.nan]
        x2nan = x2 + [np.nan]

        # test nan_policy="propagate"
        res_actual = stats.kendalltau(x1nan, x2nan,
                                      method=method, alternative=alternative)
        res_expected = (np.nan, np.nan)
        assert_allclose(res_actual, res_expected)

        # test nan_policy="omit"
        res_actual = stats.kendalltau(x1nan, x2nan, nan_policy='omit',
                                      method=method, alternative=alternative)
        res_expected = stats.kendalltau(x1, x2, method=method,
                                        alternative=alternative)
        assert_allclose(res_actual, res_expected)

        # test nan_policy="raise"
        message = 'The input contains nan values'
        with pytest.raises(ValueError, match=message):
            stats.kendalltau(x1nan, x2nan, nan_policy='raise',
                             method=method, alternative=alternative)

        # test invalid nan_policy
        message = "nan_policy must be one of..."
        with pytest.raises(ValueError, match=message):
            stats.kendalltau(x1nan, x2nan, nan_policy='ekki-ekki',
                             method=method, alternative=alternative)


def test_weightedtau():
    x = [12, 2, 1, 12, 2]
    y = [1, 4, 7, 1, 0]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(x, y, additive=False)
    assert_approx_equal(tau, -0.62205716951801038)
    assert_equal(np.nan, p_value)
    # This must be exactly Kendall's tau
    tau, p_value = stats.weightedtau(x, y, weigher=lambda x: 1)
    assert_approx_equal(tau, -0.47140452079103173)
    assert_equal(np.nan, p_value)

    # test for namedtuple attribute results
    res = stats.weightedtau(x, y)
    attributes = ('correlation', 'pvalue')
    check_named_results(res, attributes)
    assert_equal(res.correlation, res.statistic)

    # Asymmetric, ranked version
    tau, p_value = stats.weightedtau(x, y, rank=None)
    assert_approx_equal(tau, -0.4157652301037516)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(y, x, rank=None)
    assert_approx_equal(tau, -0.7181341329699029)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(x, y, rank=None, additive=False)
    assert_approx_equal(tau, -0.40644850966246893)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(y, x, rank=None, additive=False)
    assert_approx_equal(tau, -0.83766582937355172)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(x, y, rank=False)
    assert_approx_equal(tau, -0.51604397940261848)
    assert_equal(np.nan, p_value)
    # This must be exactly Kendall's tau
    tau, p_value = stats.weightedtau(x, y, rank=True, weigher=lambda x: 1)
    assert_approx_equal(tau, -0.47140452079103173)
    assert_equal(np.nan, p_value)
    tau, p_value = stats.weightedtau(y, x, rank=True, weigher=lambda x: 1)
    assert_approx_equal(tau, -0.47140452079103173)
    assert_equal(np.nan, p_value)
    # Test argument conversion
    tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), y)
    assert_approx_equal(tau, -0.56694968153682723)
    tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.int16), y)
    assert_approx_equal(tau, -0.56694968153682723)
    tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64),
                                     np.asarray(y, dtype=np.float64))
    assert_approx_equal(tau, -0.56694968153682723)
    # All ties
    with pytest.warns(SmallSampleWarning, match="One or more sample..."):
        tau, p_value = stats.weightedtau([], [])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)
    with pytest.warns(SmallSampleWarning, match="One or more sample..."):
        tau, p_value = stats.weightedtau([0], [0])
    assert_equal(np.nan, tau)
    assert_equal(np.nan, p_value)
    # Size mismatches
    assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1, 2])
    assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1], [0, 1, 2])
    # NaNs
    x = [12, 2, 1, 12, 2]
    y = [1, 4, 7, 1, np.nan]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    x = [12, 2, np.nan, 12, 2]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    # NaNs when the dtype of x and y are all np.float64
    x = [12.0, 2.0, 1.0, 12.0, 2.0]
    y = [1.0, 4.0, 7.0, 1.0, np.nan]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    x = [12.0, 2.0, np.nan, 12.0, 2.0]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.56694968153682723)
    # NaNs when there are more than one NaN in x or y
    x = [12.0, 2.0, 1.0, 12.0, 1.0]
    y = [1.0, 4.0, 7.0, 1.0, 1.0]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.6615242347139803)
    x = [12.0, 2.0, np.nan, 12.0, np.nan]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.6615242347139803)
    y = [np.nan, 4.0, 7.0, np.nan, np.nan]
    tau, p_value = stats.weightedtau(x, y)
    assert_approx_equal(tau, -0.6615242347139803)


def test_segfault_issue_9710():
    # https://github.com/scipy/scipy/issues/9710
    # This test was created to check segfault
    # In issue SEGFAULT only repros in optimized builds after calling the function twice
    message = "One or more sample arguments is too small"
    with pytest.warns(SmallSampleWarning, match=message):
        stats.weightedtau([1], [1.0])
        stats.weightedtau([1], [1.0])
        # The code below also caused SEGFAULT
        stats.weightedtau([np.nan], [52])


def test_kendall_tau_large():
    n = 172
    # Test omit policy
    x = np.arange(n + 1).astype(float)
    y = np.arange(n + 1).astype(float)
    y[-1] = np.nan
    _, pval = stats.kendalltau(x, y, method='exact', nan_policy='omit')
    assert_equal(pval, 0.0)


def test_weightedtau_vs_quadratic():
    # Trivial quadratic implementation, all parameters mandatory
    def wkq(x, y, rank, weigher, add):
        tot = conc = disc = u = v = 0
        for (i, j) in product(range(len(x)), range(len(x))):
            w = weigher(rank[i]) + weigher(rank[j]) if add \
                else weigher(rank[i]) * weigher(rank[j])
            tot += w
            if x[i] == x[j]:
                u += w
            if y[i] == y[j]:
                v += w
            if x[i] < x[j] and y[i] < y[j] or x[i] > x[j] and y[i] > y[j]:
                conc += w
            elif x[i] < x[j] and y[i] > y[j] or x[i] > x[j] and y[i] < y[j]:
                disc += w
        return (conc - disc) / np.sqrt(tot - u) / np.sqrt(tot - v)

    def weigher(x):
        return 1. / (x + 1)

    rng = np.random.default_rng(42)
    for s in range(3,10):
        a = []
        # Generate rankings with ties
        for i in range(s):
            a += [i]*i
        b = list(a)
        rng.shuffle(a)
        rng.shuffle(b)
        # First pass: use element indices as ranks
        rank = np.arange(len(a), dtype=np.intp)
        for _ in range(2):
            for add in [True, False]:
                expected = wkq(a, b, rank, weigher, add)
                actual = stats.weightedtau(a, b, rank, weigher, add).statistic
                assert_approx_equal(expected, actual)
            # Second pass: use a random rank
            rng.shuffle(rank)


class TestRegression:
    def test_linregressBIGX(self):
        # W.II.F.  Regress BIG on X.
        result = stats.linregress(X, BIG)
        assert_almost_equal(result.intercept, 99999990)
        assert_almost_equal(result.rvalue, 1.0)
        # The uncertainty ought to be almost zero
        # since all points lie on a line
        assert_almost_equal(result.stderr, 0.0)
        assert_almost_equal(result.intercept_stderr, 0.0)

    def test_regressXX(self):
        # W.IV.B.  Regress X on X.
        # The constant should be exactly 0 and the regression coefficient
        # should be 1.  This is a perfectly valid regression and the
        # program should not complain.
        result = stats.linregress(X, X)
        assert_almost_equal(result.intercept, 0.0)
        assert_almost_equal(result.rvalue, 1.0)
        # The uncertainly on regression through two points ought to be 0
        assert_almost_equal(result.stderr, 0.0)
        assert_almost_equal(result.intercept_stderr, 0.0)

        # W.IV.C. Regress X on BIG and LITTLE (two predictors).  The program
        # should tell you that this model is "singular" because BIG and
        # LITTLE are linear combinations of each other.  Cryptic error
        # messages are unacceptable here.  Singularity is the most
        # fundamental regression error.
        #
        # Need to figure out how to handle multiple linear regression.
        # This is not obvious

    def test_regressZEROX(self):
        # W.IV.D. Regress ZERO on X.
        # The program should inform you that ZERO has no variance or it should
        # go ahead and compute the regression and report a correlation and
        # total sum of squares of exactly 0.
        result = stats.linregress(X, ZERO)
        assert_almost_equal(result.intercept, 0.0)
        with pytest.warns(stats.ConstantInputWarning, match="An input array..."):
            ref_rvalue = stats.pearsonr(X, ZERO).statistic
        assert_almost_equal(result.rvalue, ref_rvalue)

    def test_regress_simple(self):
        # Regress a line with sinusoidal noise.
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))

        result = stats.linregress(x, y)
        lr = LinregressResult
        assert_(isinstance(result, lr))
        assert_almost_equal(result.stderr, 2.3957814497838803e-3)

    def test_regress_alternative(self):
        # test alternative parameter
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10  # slope is greater than zero
        y += np.sin(np.linspace(0, 20, 100))

        with pytest.raises(ValueError, match="`alternative` must be 'less'..."):
            stats.linregress(x, y, alternative="ekki-ekki")

        res1 = stats.linregress(x, y, alternative="two-sided")

        # slope is greater than zero, so "less" p-value should be large
        res2 = stats.linregress(x, y, alternative="less")
        assert_allclose(res2.pvalue, 1 - (res1.pvalue / 2))

        # slope is greater than zero, so "greater" p-value should be small
        res3 = stats.linregress(x, y, alternative="greater")
        assert_allclose(res3.pvalue, res1.pvalue / 2)

        assert res1.rvalue == res2.rvalue == res3.rvalue

    def test_regress_against_R(self):
        # test against R `lm`
        # options(digits=16)
        # x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
        # y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
        # relation <- lm(y~x)
        # print(summary(relation))

        x = [151, 174, 138, 186, 128, 136, 179, 163, 152, 131]
        y = [63, 81, 56, 91, 47, 57, 76, 72, 62, 48]
        res = stats.linregress(x, y, alternative="two-sided")
        # expected values from R's `lm` above
        assert_allclose(res.slope, 0.6746104491292)
        assert_allclose(res.intercept, -38.4550870760770)
        assert_allclose(res.rvalue, np.sqrt(0.95478224775))
        assert_allclose(res.pvalue, 1.16440531074e-06)
        assert_allclose(res.stderr, 0.0519051424731)
        assert_allclose(res.intercept_stderr, 8.0490133029927)

    def test_linregress(self):
        # compared with multivariate ols with pinv
        x = np.arange(11)
        y = np.arange(5, 16)
        y[[(1), (-2)]] -= 1
        y[[(0), (-1)]] += 1

        result = stats.linregress(x, y)

        # This test used to use 'assert_array_almost_equal' but its
        # formulation got confusing since LinregressResult became
        # _lib._bunch._make_tuple_bunch instead of namedtuple
        # (for backwards compatibility, see PR #12983)
        def assert_ae(x, y):
            return assert_almost_equal(x, y, decimal=14)
        assert_ae(result.slope, 1.0)
        assert_ae(result.intercept, 5.0)
        assert_ae(result.rvalue, 0.98229948625750)
        assert_ae(result.pvalue, 7.45259691e-008)
        assert_ae(result.stderr, 0.063564172616372733)
        assert_ae(result.intercept_stderr, 0.37605071654517686)

    def test_regress_simple_negative_cor(self):
        # If the slope of the regression is negative the factor R tend
        # to -1 not 1.  Sometimes rounding errors makes it < -1
        # leading to stderr being NaN.
        a, n = 1e-71, 100000
        x = np.linspace(a, 2 * a, n)
        y = np.linspace(2 * a, a, n)
        result = stats.linregress(x, y)

        # Make sure propagated numerical errors
        # did not bring rvalue below -1 (or were coerced)
        assert_(result.rvalue >= -1)
        assert_almost_equal(result.rvalue, -1)

        # slope and intercept stderror should stay numeric
        assert_(not np.isnan(result.stderr))
        assert_(not np.isnan(result.intercept_stderr))

    def test_linregress_result_attributes(self):
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))
        result = stats.linregress(x, y)

        # Result is of a correct class
        lr = LinregressResult
        assert_(isinstance(result, lr))

        # LinregressResult elements have correct names
        attributes = ('slope', 'intercept', 'rvalue', 'pvalue', 'stderr')
        check_named_results(result, attributes)
        # Also check that the extra attribute (intercept_stderr) is present
        assert 'intercept_stderr' in dir(result)

    def test_regress_two_inputs(self):
        # Regress a simple line formed by two points.
        x = np.arange(2)
        y = np.arange(3, 5)
        result = stats.linregress(x, y)

        # Non-horizontal line
        assert_almost_equal(result.pvalue, 0.0)

        # Zero error through two points
        assert_almost_equal(result.stderr, 0.0)
        assert_almost_equal(result.intercept_stderr, 0.0)

    def test_regress_two_inputs_horizontal_line(self):
        # Regress a horizontal line formed by two points.
        x = np.arange(2)
        y = np.ones(2)
        result = stats.linregress(x, y)

        # Horizontal line
        assert_almost_equal(result.pvalue, 1.0)

        # Zero error through two points
        assert_almost_equal(result.stderr, 0.0)
        assert_almost_equal(result.intercept_stderr, 0.0)

    def test_nist_norris(self):
        # If this causes a lint failure in the future, please note the history of
        # requests to allow extra whitespace in table formatting (e.g. gh-12367).
        # Also see https://github.com/scipy/scipy/wiki/Why-do-we-not-use-an-auto%E2%80%90formatter%3F  # noqa: E501
        x = [  0.2, 337.4, 118.2, 884.6, 10.1,  226.5,
             666.3, 996.3, 448.6, 777.0, 558.2,   0.4,
               0.6, 775.5, 666.9, 338.0, 447.5,  11.6,
             556.0, 228.1, 995.8, 887.6, 120.2,   0.3,
               0.3, 556.8, 339.1, 887.2, 999.0, 779.0,
              11.1, 118.3, 229.2, 669.1, 448.9,   0.5]

        y = [  0.1, 338.8, 118.1, 888.0, 9.2,   228.1,
             668.5, 998.5, 449.1, 778.9, 559.2,   0.3,
               0.1, 778.1, 668.8, 339.3, 448.9,  10.8,
             557.7, 228.3, 998.0, 888.8, 119.6,   0.3,
               0.6, 557.6, 339.3, 888.0, 998.5, 778.9,
              10.2, 117.6, 228.9, 668.4, 449.2,   0.2]

        result = stats.linregress(x, y)

        assert_almost_equal(result.slope, 1.00211681802045)
        assert_almost_equal(result.intercept, -0.262323073774029)
        assert_almost_equal(result.rvalue**2, 0.999993745883712)
        assert_almost_equal(result.pvalue, 0.0)
        assert_almost_equal(result.stderr, 0.00042979684820)
        assert_almost_equal(result.intercept_stderr, 0.23281823430153)

    def test_compare_to_polyfit(self):
        x = np.linspace(0, 100, 100)
        y = 0.2 * np.linspace(0, 100, 100) + 10
        y += np.sin(np.linspace(0, 20, 100))
        result = stats.linregress(x, y)
        poly = np.polyfit(x, y, 1)  # Fit 1st degree polynomial

        # Make sure linear regression slope and intercept
        # match with results from numpy polyfit
        assert_almost_equal(result.slope, poly[0])
        assert_almost_equal(result.intercept, poly[1])

    def test_empty_input(self):
        with pytest.warns(SmallSampleWarning, match="One or more sample..."):
            res = stats.linregress([], [])
            assert np.all(np.isnan(res))

    def test_nan_input(self):
        x = np.arange(10.)
        x[9] = np.nan

        with np.errstate(invalid="ignore"):
            result = stats.linregress(x, x)

        # Make sure the result still comes back as `LinregressResult`
        lr = LinregressResult
        assert_(isinstance(result, lr))
        assert_array_equal(result, (np.nan,)*5)
        assert_equal(result.intercept_stderr, np.nan)

    def test_identical_x(self):
        rng = np.random.default_rng(7872425088)
        x = np.zeros(10)
        y = rng.random(10)
        msg = "Cannot calculate a linear regression"
        with assert_raises(ValueError, match=msg):
            stats.linregress(x, y)


def test_theilslopes():
    # Basic slope test.
    slope, intercept, lower, upper = stats.theilslopes([0,1,1])
    assert_almost_equal(slope, 0.5)
    assert_almost_equal(intercept, 0.5)

    msg = ("method must be either 'joint' or 'separate'."
           "'joint_separate' is invalid.")
    with pytest.raises(ValueError, match=msg):
        stats.theilslopes([0, 1, 1], method='joint_separate')

    slope, intercept, lower, upper = stats.theilslopes([0, 1, 1],
                                                       method='joint')
    assert_almost_equal(slope, 0.5)
    assert_almost_equal(intercept, 0.0)

    # Test of confidence intervals.
    x = [1, 2, 3, 4, 10, 12, 18]
    y = [9, 15, 19, 20, 45, 55, 78]
    slope, intercept, lower, upper = stats.theilslopes(y, x, 0.07,
                                                       method='separate')
    assert_almost_equal(slope, 4)
    assert_almost_equal(intercept, 4.0)
    assert_almost_equal(upper, 4.38, decimal=2)
    assert_almost_equal(lower, 3.71, decimal=2)

    slope, intercept, lower, upper = stats.theilslopes(y, x, 0.07,
                                                       method='joint')
    assert_almost_equal(slope, 4)
    assert_almost_equal(intercept, 6.0)
    assert_almost_equal(upper, 4.38, decimal=2)
    assert_almost_equal(lower, 3.71, decimal=2)


def test_cumfreq():
    x = [1, 4, 2, 1, 3, 1]
    cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4)
    assert_array_almost_equal(cumfreqs, np.array([3., 4., 5., 6.]))
    cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(
        x, numbins=4, defaultreallimits=(1.5, 5))
    assert_(extrapoints == 3)

    # test for namedtuple attribute results
    attributes = ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')
    res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
    check_named_results(res, attributes)


def test_relfreq():
    a = np.array([1, 4, 2, 1, 3, 1])
    relfreqs, lowlim, binsize, extrapoints = stats.relfreq(a, numbins=4)
    assert_array_almost_equal(relfreqs,
                              array([0.5, 0.16666667, 0.16666667, 0.16666667]))

    # test for namedtuple attribute results
    attributes = ('frequency', 'lowerlimit', 'binsize', 'extrapoints')
    res = stats.relfreq(a, numbins=4)
    check_named_results(res, attributes)

    # check array_like input is accepted
    relfreqs2, lowlim, binsize, extrapoints = stats.relfreq([1, 4, 2, 1, 3, 1],
                                                            numbins=4)
    assert_array_almost_equal(relfreqs, relfreqs2)


class TestScoreatpercentile:
    def setup_method(self):
        self.a1 = [3, 4, 5, 10, -3, -5, 6]
        self.a2 = [3, -6, -2, 8, 7, 4, 2, 1]
        self.a3 = [3., 4, 5, 10, -3, -5, -6, 7.0]

    def test_basic(self):
        x = arange(8) * 0.5
        assert_equal(stats.scoreatpercentile(x, 0), 0.)
        assert_equal(stats.scoreatpercentile(x, 100), 3.5)
        assert_equal(stats.scoreatpercentile(x, 50), 1.75)

    def test_fraction(self):
        scoreatperc = stats.scoreatpercentile

        # Test defaults
        assert_equal(scoreatperc(list(range(10)), 50), 4.5)
        assert_equal(scoreatperc(list(range(10)), 50, (2,7)), 4.5)
        assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8)), 4.5)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10,100)), 55)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10)), 5.5)

        # explicitly specify interpolation_method 'fraction' (the default)
        assert_equal(scoreatperc(list(range(10)), 50, interpolation_method='fraction'),
                     4.5)
        assert_equal(scoreatperc(list(range(10)), 50, limit=(2, 7),
                                 interpolation_method='fraction'),
                     4.5)
        assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8),
                                 interpolation_method='fraction'),
                     4.5)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10, 100),
                                 interpolation_method='fraction'),
                     55)
        assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10),
                                 interpolation_method='fraction'),
                     5.5)

    def test_lower_higher(self):
        scoreatperc = stats.scoreatpercentile

        # interpolation_method 'lower'/'higher'
        assert_equal(scoreatperc(list(range(10)), 50,
                                 interpolation_method='lower'), 4)
        assert_equal(scoreatperc(list(range(10)), 50,
                                 interpolation_method='higher'), 5)
        assert_equal(scoreatperc(list(range(10)), 50, (2,7),
                                 interpolation_method='lower'), 4)
        assert_equal(scoreatperc(list(range(10)), 50, limit=(2,7),
                                 interpolation_method='higher'), 5)
        assert_equal(scoreatperc(list(range(100)), 50, (1,8),
                                 interpolation_method='lower'), 4)
        assert_equal(scoreatperc(list(range(100)), 50, (1,8),
                                 interpolation_method='higher'), 5)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (10, 100),
                                 interpolation_method='lower'), 10)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(10, 100),
                                 interpolation_method='higher'), 100)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (1, 10),
                                 interpolation_method='lower'), 1)
        assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(1, 10),
                                 interpolation_method='higher'), 10)

    def test_sequence_per(self):
        x = arange(8) * 0.5
        expected = np.array([0, 3.5, 1.75])
        res = stats.scoreatpercentile(x, [0, 100, 50])
        assert_allclose(res, expected)
        assert_(isinstance(res, np.ndarray))
        # Test with ndarray.  Regression test for gh-2861
        assert_allclose(stats.scoreatpercentile(x, np.array([0, 100, 50])),
                        expected)
        # Also test combination of 2-D array, axis not None and array-like per
        res2 = stats.scoreatpercentile(np.arange(12).reshape((3,4)),
                                       np.array([0, 1, 100, 100]), axis=1)
        expected2 = array([[0, 4, 8],
                           [0.03, 4.03, 8.03],
                           [3, 7, 11],
                           [3, 7, 11]])
        assert_allclose(res2, expected2)

    def test_axis(self):
        scoreatperc = stats.scoreatpercentile
        x = arange(12).reshape(3, 4)

        assert_equal(scoreatperc(x, (25, 50, 100)), [2.75, 5.5, 11.0])

        r0 = [[2, 3, 4, 5], [4, 5, 6, 7], [8, 9, 10, 11]]
        assert_equal(scoreatperc(x, (25, 50, 100), axis=0), r0)

        r1 = [[0.75, 4.75, 8.75], [1.5, 5.5, 9.5], [3, 7, 11]]
        assert_equal(scoreatperc(x, (25, 50, 100), axis=1), r1)

        x = array([[1, 1, 1],
                   [1, 1, 1],
                   [4, 4, 3],
                   [1, 1, 1],
                   [1, 1, 1]])
        score = stats.scoreatpercentile(x, 50)
        assert_equal(score.shape, ())
        assert_equal(score, 1.0)
        score = stats.scoreatpercentile(x, 50, axis=0)
        assert_equal(score.shape, (3,))
        assert_equal(score, [1, 1, 1])

    def test_exception(self):
        assert_raises(ValueError, stats.scoreatpercentile, [1, 2], 56,
                      interpolation_method='foobar')
        assert_raises(ValueError, stats.scoreatpercentile, [1], 101)
        assert_raises(ValueError, stats.scoreatpercentile, [1], -1)

    def test_empty(self):
        assert_equal(stats.scoreatpercentile([], 50), np.nan)
        assert_equal(stats.scoreatpercentile(np.array([[], []]), 50), np.nan)
        assert_equal(stats.scoreatpercentile([], [50, 99]), [np.nan, np.nan])


@make_xp_test_case(stats.mode)
class TestMode:

    def test_empty(self, xp):
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            vals, counts = stats.mode(xp.asarray([]))
        xp_assert_equal(vals, xp.asarray(xp.nan))
        xp_assert_equal(counts, xp.asarray(0.))

    def test_scalar(self):
        vals, counts = stats.mode(4.)
        assert_equal(vals, np.array([4.]))
        assert_equal(counts, np.array([1]))

    def test_basic(self, xp):
        data1 = xp.asarray([3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6])
        vals = stats.mode(data1)
        xp_assert_equal(vals[0], xp.asarray(6))
        xp_assert_equal(vals[1], xp.asarray(3))

    def test_axes_keepdims(self, xp):
        data1 = [10, 10, 30, 40]
        data2 = [10, 10, 10, 10]
        data3 = [20, 10, 20, 20]
        data4 = [30, 30, 30, 30]
        data5 = [40, 30, 30, 30]
        arr = xp.asarray([data1, data2, data3, data4, data5])

        vals = stats.mode(arr, axis=None, keepdims=True)
        xp_assert_equal(vals[0], xp.asarray([[30]]))
        xp_assert_equal(vals[1], xp.asarray([[8]]))

        vals = stats.mode(arr, axis=0, keepdims=True)
        xp_assert_equal(vals[0], xp.asarray([[10, 10, 30, 30]]))
        xp_assert_equal(vals[1], xp.asarray([[2, 3, 3, 2]]))

        vals = stats.mode(arr, axis=1, keepdims=True)
        xp_assert_equal(vals[0], xp.asarray([[10], [10], [20], [30], [30]]))
        xp_assert_equal(vals[1], xp.asarray([[2], [4], [3], [4], [3]]))

    def test_axes(self, xp):
        data1 = [10, 10, 30, 40]
        data2 = [10, 10, 10, 10]
        data3 = [20, 10, 20, 20]
        data4 = [30, 30, 30, 30]
        data5 = [40, 30, 30, 30]
        arr = xp.asarray([data1, data2, data3, data4, data5])

        vals = stats.mode(arr, axis=None)
        xp_assert_equal(vals[0], xp.asarray(30))
        xp_assert_equal(vals[1], xp.asarray(8))

        vals = stats.mode(arr, axis=0)
        xp_assert_equal(vals[0], xp.asarray([10, 10, 30, 30]))
        xp_assert_equal(vals[1], xp.asarray([2, 3, 3, 2]))

        vals = stats.mode(arr, axis=1)
        xp_assert_equal(vals[0], xp.asarray([10, 10, 20, 30, 30]))
        xp_assert_equal(vals[1], xp.asarray([2, 4, 3, 4, 3]))

    @pytest.mark.parametrize('axis', range(-4, 0))
    def test_negative_axes_gh_15375(self, axis, xp):
        rng = np.random.default_rng(7090348401)
        a = xp.asarray(rng.random((10, 11, 12, 13)))
        res0 = stats.mode(a, axis=a.ndim+axis)
        res1 = stats.mode(a, axis=axis)
        xp_assert_equal(res0.mode, res1.mode)
        xp_assert_equal(res0.count, res1.count)

    def test_mode_result_attributes(self, xp):
        data1 = xp.asarray([3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6])
        data2 = xp.asarray([])
        actual = stats.mode(data1)
        attributes = ('mode', 'count')
        check_named_results(actual, attributes, xp=xp)
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            actual2 = stats.mode(data2)
        check_named_results(actual2, attributes, xp=xp)

    def test_nan_propagate(self, xp):
        data1 = xp.asarray([3, np.nan, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6])
        actual = stats.mode(data1)
        xp_assert_equal(actual[0], xp.asarray(6, dtype=data1.dtype))
        xp_assert_equal(actual[1], xp.asarray(3))

    @skip_xp_backends(eager_only=True, reason="lazy arrays don't do 'raise'.")
    def test_nan_omit(self, xp):
        data1 = xp.asarray([3, np.nan, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6])
        res = stats.mode(data1, nan_policy='omit')
        xp_assert_equal(res.mode, xp.asarray(6.))
        xp_assert_equal(res.count, xp.asarray(3))
        assert_raises(ValueError, stats.mode, data1, nan_policy='raise')
        assert_raises(ValueError, stats.mode, data1, nan_policy='foobar')

    @skip_xp_backends(eager_only=True, reason="lazy arrays don't do 'omit'.")
    @pytest.mark.parametrize("data", [
        [3, 5, 1, 1, 3.],
        [3, np.nan, 5, 1, 1, 3],
        [3, 5, 1.],
        [3, np.nan, 5, 1],
    ])
    @pytest.mark.parametrize('keepdims', [False, True])
    def test_smallest_equal(self, data, keepdims, xp):
        result = stats.mode(xp.asarray(data), nan_policy='omit', keepdims=keepdims)
        if keepdims:
            xp_assert_equal(result[0][0], xp.asarray(1.))
        else:
            xp_assert_equal(result[0], xp.asarray(1.))

    @pytest.mark.parametrize('axis', range(-3, 3))
    def test_mode_shape_gh_9955(self, axis, xp):
        rng = np.random.default_rng(984213899)
        a = xp.asarray(rng.uniform(size=(3, 4, 5)))
        res = stats.mode(a, axis=axis, keepdims=False)
        reference_shape = list(a.shape)
        reference_shape.pop(axis)
        np.testing.assert_array_equal(res.mode.shape, reference_shape)
        np.testing.assert_array_equal(res.count.shape, reference_shape)

    def test_nan_policy_propagate_gh_9815(self, xp):
        # mode should treat np.nan as it would any other object when
        # nan_policy='propagate'
        a = xp.asarray([2, np.nan, 1, np.nan])
        res = stats.mode(a)
        assert xp.isnan(res.mode) and res.count == 2

    def test_keepdims_empty(self, xp):
        # test empty arrays
        a = xp.zeros((1, 2, 3, 0))

        res = stats.mode(a, axis=1, keepdims=False)
        assert res.mode.shape == res.count.shape == (1, 3, 0)

        res = stats.mode(a, axis=1, keepdims=True)
        assert res.mode.shape == res.count.shape == (1, 1, 3, 0)

    def test_keepdims_nonempty(selfself, xp):
        # test nan_policy='propagate'
        a = xp.asarray([[1, 3, 3, np.nan], [1, 1, np.nan, 1]])

        res = stats.mode(a, axis=1, keepdims=False)
        xp_assert_equal(res.mode, xp.asarray([3., 1.]))
        xp_assert_equal(res.count, xp.asarray([2, 3]))

        res = stats.mode(a, axis=1, keepdims=True)
        xp_assert_equal(res.mode, xp.asarray([[3.], [1.]]))
        xp_assert_equal(res.count, xp.asarray([[2], [3]]))

        a = xp.asarray(a)
        res = stats.mode(a, axis=None, keepdims=False)
        ref = stats.mode(xp_ravel(a), keepdims=False)
        xp_assert_equal(res.mode, ref.mode)
        xp_assert_equal(res.count, ref.count)
        assert res.mode.shape == ref.mode.shape == ()

        res = stats.mode(a, axis=None, keepdims=True)
        ref = stats.mode(xp_ravel(a), keepdims=True)
        xp_assert_equal(xp_ravel(res.mode), xp_ravel(ref.mode))
        assert res.mode.shape == (1, 1)
        xp_assert_equal(xp_ravel(res.count), xp_ravel(ref.count))
        assert res.count.shape == (1, 1)

    def test_keepdims_nan_omit(self):
        # test nan_policy='omit'
        a = [[1, np.nan, np.nan, np.nan, 1],
             [np.nan, np.nan, np.nan, np.nan, 2],
             [1, 2, np.nan, 5, 5]]

        res = stats.mode(a, axis=1, keepdims=False, nan_policy='omit')
        assert_array_equal(res.mode, [1, 2, 5])
        assert_array_equal(res.count, [2, 1, 2])

        res = stats.mode(a, axis=1, keepdims=True, nan_policy='omit')
        assert_array_equal(res.mode, [[1], [2], [5]])
        assert_array_equal(res.count, [[2], [1], [2]])

        a = np.array(a)
        res = stats.mode(a, axis=None, keepdims=False, nan_policy='omit')
        ref = stats.mode(a.ravel(), keepdims=False, nan_policy='omit')
        assert_array_equal(res, ref)
        assert res.mode.shape == ref.mode.shape == ()

        res = stats.mode(a, axis=None, keepdims=True, nan_policy='omit')
        ref = stats.mode(a.ravel(), keepdims=True, nan_policy='omit')
        assert_equal(res.mode.ravel(), ref.mode.ravel())
        assert res.mode.shape == (1, 1)
        assert_equal(res.count.ravel(), ref.count.ravel())
        assert res.count.shape == (1, 1)

    @pytest.mark.parametrize("nan_policy", ['propagate', 'omit'])
    def test_gh16955(self, nan_policy):
        # Check that bug reported in gh-16955 is resolved
        shape = (4, 3)
        data = np.ones(shape)
        data[0, 0] = np.nan
        res = stats.mode(a=data, axis=1, keepdims=False, nan_policy=nan_policy)
        assert_array_equal(res.mode, [1, 1, 1, 1])
        assert_array_equal(res.count, [2, 3, 3, 3])

        # Test with input from gh-16595. Support for non-numeric input
        # was deprecated, so check for the appropriate error.
        my_dtype = np.dtype([('asdf', np.uint8), ('qwer', np.float64, (3,))])
        test = np.zeros(10, dtype=my_dtype)
        message = "Argument `a` is not....|An argument has dtype...|The DType..."
        with pytest.raises(TypeError, match=message):
            stats.mode(test, nan_policy=nan_policy)

    def test_gh9955(self):
        # The behavior of mode with empty slices (whether the input was empty
        # or all elements were omitted) was inconsistent. Test that this is
        # resolved: the mode of an empty slice is NaN and the count is zero.
        with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
            res = stats.mode([])
        ref = (np.nan, 0)
        assert_equal(res, ref)

        with pytest.warns(SmallSampleWarning, match=too_small_1d_omit):
            res = stats.mode([np.nan], nan_policy='omit')
        assert_equal(res, ref)

        a = [[10., 20., 20.], [np.nan, np.nan, np.nan]]
        with pytest.warns(SmallSampleWarning, match=too_small_nd_omit):
            res = stats.mode(a, axis=1, nan_policy='omit')
        ref = ([20, np.nan], [2, 0])
        assert_equal(res, ref)

        res = stats.mode(a, axis=1, nan_policy='propagate')
        ref = ([20, np.nan], [2, 3])
        assert_equal(res, ref)

        z = np.array([[], []])
        with pytest.warns(SmallSampleWarning, match=too_small_nd_not_omit):
            res = stats.mode(z, axis=1)
        ref = ([np.nan, np.nan], [0, 0])
        assert_equal(res, ref)

    @pytest.mark.filterwarnings('ignore::RuntimeWarning')  # np.mean warns
    @pytest.mark.parametrize('z', [np.empty((0, 1, 2)), np.empty((1, 1, 2))])
    def test_gh17214(self, z, xp):
        z = xp.asarray(z)
        if z.size == 0:
            with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
                res = stats.mode(z, axis=None, keepdims=True)
        else:
            res = stats.mode(z, axis=None, keepdims=True)
        ref = xp.mean(z, axis=None, keepdims=True)
        assert res[0].shape == res[1].shape == ref.shape == (1, 1, 1)

    def test_raise_non_numeric_gh18254(self):

        class ArrLike:
            def __init__(self, x):
                self._x = x

            def __array__(self, dtype=None, copy=None):
                return self._x.astype(object)

        message = ("...only boolean and numerical dtypes..." if SCIPY_ARRAY_API
                   else "Cannot interpret...")

        with pytest.raises(TypeError, match=message):
            stats.mode(ArrLike(np.arange(3)))

        message = ("...only boolean and numerical dtypes..." if SCIPY_ARRAY_API
                   else "Argument `a` is not recognized as numeric.")
        with pytest.raises(TypeError, match=message):
            stats.mode(np.arange(3, dtype=object))


@make_xp_test_case(stats.sem)
class TestSEM:

    testcase = [1., 2., 3., 4.]
    scalar_testcase = 4.

    @pytest.mark.filterwarnings("ignore:invalid value encountered in divide")
    def test_sem_scalar(self, xp):
        # This is not in R, so used:
        #     sqrt(var(testcase)*3/4)/sqrt(3)

        # y = stats.sem(self.shoes[0])
        # assert_approx_equal(y,0.775177399)
        scalar_testcase = xp.asarray(self.scalar_testcase)[()]
        if is_numpy(xp):
            with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
                y = stats.sem(scalar_testcase)
        else:
            # Other array types can emit a variety of warnings.
            with warnings.catch_warnings():
                warnings.simplefilter("ignore", UserWarning)
                warnings.simplefilter("ignore", RuntimeWarning)
                y = stats.sem(scalar_testcase)
        assert xp.isnan(y)

    def test_sem(self, xp):
        testcase = xp.asarray(self.testcase)
        y = stats.sem(testcase)
        xp_assert_close(y, xp.asarray(0.6454972244))
        n = len(self.testcase)
        xp_assert_close(stats.sem(testcase, ddof=0) * (n/(n-2))**0.5,
                        stats.sem(testcase, ddof=2))

        x = xp.arange(10.)
        x = xp.where(x == 9, xp.nan, x)
        xp_assert_equal(stats.sem(x), xp.asarray(xp.nan))

    @skip_xp_backends(np_only=True,
                      reason='`nan_policy` only supports NumPy backend')
    def test_sem_nan_policy(self, xp):
        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.sem(x, nan_policy='omit'), 0.9128709291752769)
        assert_raises(ValueError, stats.sem, x, nan_policy='raise')
        assert_raises(ValueError, stats.sem, x, nan_policy='foobar')


@make_xp_test_case(stats.zmap)
class TestZmap:

    @pytest.mark.parametrize(
        'x, y',
        [([1., 2., 3., 4.], [1., 2., 3., 4.]),
         ([1., 2., 3.], [0., 1., 2., 3., 4.])]
    )
    def test_zmap(self, x, y, xp):
        # For these simple cases, calculate the expected result directly
        # by using the formula for the z-score.
        x, y = xp.asarray(x), xp.asarray(y)
        expected = (x - xp.mean(y)) / xp.std(y, correction=0)
        z = stats.zmap(x, y)
        xp_assert_close(z, expected)

    def test_zmap_axis(self, xp):
        # Test use of 'axis' keyword in zmap.
        x = xp.asarray([[0.0, 0.0, 1.0, 1.0],
                        [1.0, 1.0, 1.0, 2.0],
                        [2.0, 0.0, 2.0, 0.0]])

        t1 = 1.0/(2.0/3)**0.5
        t2 = 3.**0.5/3
        t3 = 2.**0.5

        z0 = stats.zmap(x, x, axis=0)
        z1 = stats.zmap(x, x, axis=1)

        z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
                       [0.0, t3, -t3/2, t1],
                       [t1, -t3/2, t3, -t1]]
        z1_expected = [[-1.0, -1.0, 1.0, 1.0],
                       [-t2, -t2, -t2, 3.**0.5],
                       [1.0, -1.0, 1.0, -1.0]]
        z0_expected = xp.asarray(z0_expected)
        z1_expected = xp.asarray(z1_expected)

        xp_assert_close(z0, z0_expected)
        xp_assert_close(z1, z1_expected)

    def test_zmap_ddof(self, xp):
        # Test use of 'ddof' keyword in zmap.
        x = xp.asarray([[0.0, 0.0, 1.0, 1.0],
                        [0.0, 1.0, 2.0, 3.0]])

        z = stats.zmap(x, x, axis=1, ddof=1)

        z0_expected = xp.asarray([-0.5, -0.5, 0.5, 0.5])/(1.0/3**0.5)
        z1_expected = xp.asarray([-1.5, -0.5, 0.5, 1.5])/(5./3)**0.5
        xp_assert_close(z[0, :], z0_expected)
        xp_assert_close(z[1, :], z1_expected)

    @pytest.mark.parametrize('ddof', [0, 2])
    def test_zmap_nan_policy_omit(self, ddof, xp):
        # nans in `scores` are propagated, regardless of `nan_policy`.
        # `nan_policy` only affects how nans in `compare` are handled.
        scores = xp.asarray([-3, -1, 2, np.nan])
        compare = xp.asarray([-8, -3, 2, 7, 12, np.nan])
        z = stats.zmap(scores, compare, ddof=ddof, nan_policy='omit')
        # exclude nans from compare, don't use isnan + mask since that messes up
        # dask
        ref = stats.zmap(scores, compare[:5], ddof=ddof)
        xp_assert_close(z, ref)

    @pytest.mark.parametrize('ddof', [0, 2])
    def test_zmap_nan_policy_omit_with_axis(self, ddof, xp):
        scores = xp.reshape(xp.arange(-5.0, 9.0), (2, -1))
        compare = np.reshape(np.linspace(-8, 6, 24), (2, -1))
        compare[0, 4] = np.nan
        compare[0, 6] = np.nan
        compare[1, 1] = np.nan
        # convert from numpy since some libraries like dask
        # can't handle the data-dependent shapes from the isnan masking
        compare_0_notna = xp.asarray(compare[0, :][~np.isnan(compare[0, :])])
        compare_1_notna = xp.asarray(compare[1, :][~np.isnan(compare[1, :])])
        compare = xp.asarray(compare)

        z = stats.zmap(scores, compare, nan_policy='omit', axis=1, ddof=ddof)
        res0 = stats.zmap(scores[0, :], compare_0_notna,
                          ddof=ddof)
        res1 = stats.zmap(scores[1, :], compare_1_notna,
                          ddof=ddof)
        expected = xp.stack((res0, res1))
        xp_assert_close(z, expected)

    @skip_xp_backends(eager_only=True, reason="lazy arrays don't do 'raise'.")
    def test_zmap_nan_policy_raise(self, xp):
        scores = xp.asarray([1, 2, 3])
        compare = xp.asarray([-8, -3, 2, 7, 12, xp.nan])
        with pytest.raises(ValueError, match='input contains nan'):
            stats.zmap(scores, compare, nan_policy='raise')

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_degenerate_input(self, xp):
        scores = xp.arange(3)
        compare = xp.ones(3)
        ref = xp.asarray([-xp.inf, xp.nan, xp.inf])
        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            res = stats.zmap(scores, compare)
        xp_assert_equal(res, ref)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_complex_gh22404(self, xp):
        res = stats.zmap(xp.asarray([1, 2, 3, 4]), xp.asarray([1, 1j, -1, -1j]))
        ref = xp.asarray([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j])
        xp_assert_close(res, ref)


@make_xp_test_case(stats.zscore)
class TestZscore:
    def test_zscore(self, xp):
        # not in R, so tested by using:
        #    (testcase[i] - mean(testcase, axis=0)) / sqrt(var(testcase) * 3/4)
        y = stats.zscore(xp.asarray([1, 2, 3, 4]))
        desired = [-1.3416407864999, -0.44721359549996,
                   0.44721359549996, 1.3416407864999]
        xp_assert_close(y, xp.asarray(desired))

    def test_zscore_axis(self, xp):
        # Test use of 'axis' keyword in zscore.
        x = xp.asarray([[0.0, 0.0, 1.0, 1.0],
                        [1.0, 1.0, 1.0, 2.0],
                        [2.0, 0.0, 2.0, 0.0]])

        t1 = 1.0/(2.0/3)**0.5
        t2 = 3**0.5/3
        t3 = 2**0.5

        z0 = stats.zscore(x, axis=0)
        z1 = stats.zscore(x, axis=1)

        z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
                       [0.0, t3, -t3/2, t1],
                       [t1, -t3/2, t3, -t1]]
        z1_expected = [[-1.0, -1.0, 1.0, 1.0],
                       [-t2, -t2, -t2, 3**0.5],
                       [1.0, -1.0, 1.0, -1.0]]

        xp_assert_close(z0, xp.asarray(z0_expected))
        xp_assert_close(z1, xp.asarray(z1_expected))

    def test_zscore_ddof(self, xp):
        # Test use of 'ddof' keyword in zscore.
        x = xp.asarray([[0.0, 0.0, 1.0, 1.0],
                        [0.0, 1.0, 2.0, 3.0]])

        z = stats.zscore(x, axis=1, ddof=1)

        z0_expected = xp.asarray([-0.5, -0.5, 0.5, 0.5])/(1.0/3**0.5)
        z1_expected = xp.asarray([-1.5, -0.5, 0.5, 1.5])/((5./3)**0.5)
        xp_assert_close(z[0, :], z0_expected)
        xp_assert_close(z[1, :], z1_expected)

    def test_zscore_nan_propagate(self, xp):
        x = xp.asarray([1, 2, np.nan, 4, 5])
        z = stats.zscore(x, nan_policy='propagate')
        xp_assert_equal(z, xp.full(x.shape, xp.nan))

    def test_zscore_nan_omit(self, xp):
        x = xp.asarray([1, 2, xp.nan, 4, 5])

        z = stats.zscore(x, nan_policy='omit')

        expected = xp.asarray([-1.2649110640673518,
                               -0.6324555320336759,
                               xp.nan,
                               0.6324555320336759,
                               1.2649110640673518
                               ])
        xp_assert_close(z, expected)

    def test_zscore_nan_omit_with_ddof(self, xp):
        x = xp.asarray([xp.nan, 1.0, 3.0, 5.0, 7.0, 9.0])
        z = stats.zscore(x, ddof=1, nan_policy='omit')
        expected = xp.concat([xp.asarray([xp.nan]), stats.zscore(x[1:], ddof=1)])
        xp_assert_close(z, expected)

    @skip_xp_backends(eager_only=True, reason="lazy arrays don't do 'raise'.")
    def test_zscore_nan_raise(self, xp):
        x = xp.asarray([1, 2, xp.nan, 4, 5])
        with pytest.raises(ValueError, match="The input contains nan..."):
            stats.zscore(x, nan_policy='raise')

    def test_zscore_constant_input_1d(self, xp):
        x = xp.asarray([-0.087] * 3)
        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            z = stats.zscore(x)
        xp_assert_equal(z, xp.full(x.shape, xp.nan))

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_zscore_constant_input_2d(self, xp):
        x = xp.asarray([[10.0, 10.0, 10.0, 10.0],
                        [10.0, 11.0, 12.0, 13.0]])
        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            z0 = stats.zscore(x, axis=0)
        xp_assert_close(z0, xp.asarray([[xp.nan, -1.0, -1.0, -1.0],
                                        [xp.nan, 1.0, 1.0, 1.0]]))

        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            z1 = stats.zscore(x, axis=1)
        xp_assert_equal(z1, xp.stack([xp.asarray([xp.nan, xp.nan, xp.nan, xp.nan]),
                                      stats.zscore(x[1, :])]))

        z = stats.zscore(x, axis=None)
        xp_assert_equal(z, xp.reshape(stats.zscore(xp.reshape(x, (-1,))), x.shape))

        y = xp.ones((3, 6))
        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            z = stats.zscore(y, axis=None)
        xp_assert_equal(z, xp.full_like(y, xp.nan))

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_zscore_constant_input_2d_nan_policy_omit(self, xp):
        x = xp.asarray([[10.0, 10.0, 10.0, 10.0],
                        [10.0, 11.0, 12.0, xp.nan],
                        [10.0, 12.0, xp.nan, 10.0]])
        s = (3/2)**0.5
        s2 = 2**0.5

        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            z0 = stats.zscore(x, nan_policy='omit', axis=0)
        xp_assert_close(z0, xp.asarray([[xp.nan, -s, -1.0, xp.nan],
                                        [xp.nan, 0, 1.0, xp.nan],
                                        [xp.nan, s, xp.nan, xp.nan]]))

        with eager_warns(RuntimeWarning, match="Precision loss occurred...", xp=xp):
            z1 = stats.zscore(x, nan_policy='omit', axis=1)
        xp_assert_close(z1, xp.asarray([[xp.nan, xp.nan, xp.nan, xp.nan],
                                        [-s, 0, s, xp.nan],
                                        [-s2/2, s2, xp.nan, -s2/2]]))

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_zscore_2d_all_nan_row(self, xp):
        # A row is all nan, and we use axis=1.
        x = xp.asarray([[np.nan, np.nan, np.nan, np.nan],
                        [10.0, 10.0, 12.0, 12.0]])
        z = stats.zscore(x, nan_policy='omit', axis=1)
        xp_assert_close(z, xp.asarray([[np.nan, np.nan, np.nan, np.nan],
                                       [-1.0, -1.0, 1.0, 1.0]]))

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_zscore_2d_all_nan(self, xp):
        # The entire 2d array is nan, and we use axis=None.
        y = xp.full((2, 3), xp.nan)
        z = stats.zscore(y, nan_policy='omit', axis=None)
        xp_assert_equal(z, y)

    @pytest.mark.parametrize('x', [np.array([]), np.zeros((3, 0, 5))])
    def test_zscore_empty_input(self, x, xp):
        x = xp.asarray(x)
        z = stats.zscore(x)
        xp_assert_equal(z, x)

    @skip_xp_invalid_arg
    def test_zscore_masked_element_0_gh19039(self, xp):
        # zscore returned all NaNs when 0th element was masked. See gh-19039.
        rng = np.random.default_rng(8675309)
        x = rng.standard_normal(10)
        mask = np.zeros_like(x)
        y = np.ma.masked_array(x, mask)
        y.mask[0] = True

        ref = stats.zscore(x[1:])  # compute reference from non-masked elements
        assert not np.any(np.isnan(ref))
        res = stats.zscore(y)
        assert_allclose(res[1:], ref)
        res = stats.zscore(y, axis=None)
        assert_allclose(res[1:], ref)

        y[1:] = y[1]  # when non-masked elements are identical, result is nan
        with pytest.warns(RuntimeWarning, match="Precision loss occurred..."):
            res = stats.zscore(y)
        assert_equal(res[1:], np.nan)
        with pytest.warns(RuntimeWarning, match="Precision loss occurred..."):
            res = stats.zscore(y, axis=None)
        assert_equal(res[1:], np.nan)


@make_xp_test_case(stats.gzscore)
class TestGZscore:
    def test_gzscore_normal_array(self, xp):
        x = np.asarray([1, 2, 3, 4])
        z = stats.gzscore(xp.asarray(x))
        desired = np.log(x / stats.gmean(x)) / np.log(stats.gstd(x, ddof=0))
        xp_assert_close(z, xp.asarray(desired, dtype=xp.asarray(1.).dtype))

    @skip_xp_invalid_arg
    def test_gzscore_masked_array(self):
        x = np.array([1, 2, -1, 3, 4])
        mask = [0, 0, 1, 0, 0]
        mx = np.ma.masked_array(x, mask=mask)
        z = stats.gzscore(mx)
        desired = ([-1.526072095151, -0.194700599824, np.inf, 0.584101799472,
                    1.136670895503])
        desired = np.ma.masked_array(desired, mask=mask)
        assert_allclose(z.compressed(), desired.compressed())
        assert_allclose(z.mask, desired.mask)
        assert isinstance(z, np.ma.MaskedArray)


@make_xp_test_case(stats.median_abs_deviation)
class TestMedianAbsDeviation:
    def setup_method(self):
        self.dat_nan = [2.20, 2.20, 2.4, 2.4, 2.5, 2.7, 2.8, 2.9,
                        3.03, 3.03, 3.10, 3.37, 3.4, 3.4, 3.4, 3.5,
                        3.6, 3.7, 3.7, 3.7, 3.7, 3.77, 5.28, np.nan]
        self.dat = [2.20, 2.20, 2.4, 2.4, 2.5, 2.7, 2.8, 2.9, 3.03,
                    3.03, 3.10, 3.37, 3.4, 3.4, 3.4, 3.5, 3.6, 3.7,
                    3.7, 3.7, 3.7, 3.77, 5.28, 28.95]

    def test_median_abs_deviation(self, xp):
        xp_assert_close(stats.median_abs_deviation(xp.asarray(self.dat), axis=None),
                        xp.asarray(0.355))
        dat = xp.reshape(xp.asarray(self.dat), (6, 4))
        mad = stats.median_abs_deviation(dat, axis=0)
        mad_expected = xp.asarray([0.435, 0.5, 0.45, 0.4])
        xp_assert_close(mad, mad_expected)

    def test_mad_nan_omit(self, xp):
        mad = stats.median_abs_deviation(xp.asarray(self.dat_nan), nan_policy='omit')
        xp_assert_close(mad, xp.asarray(0.34))

    def test_axis_and_nan(self, xp):
        x = xp.asarray([[1.0, 2.0, 3.0, 4.0, np.nan],
                        [1.0, 4.0, 5.0, 8.0, 9.0]])
        mad = stats.median_abs_deviation(x, axis=1)
        xp_assert_close(mad, xp.asarray([np.nan, 3.0]))

    def test_nan_policy_omit_with_inf(self, xp):
        z = xp.asarray([1, 3, 4, 6, 99, np.nan, np.inf])
        mad = stats.median_abs_deviation(z, nan_policy='omit')
        xp_assert_close(mad, xp.asarray(3.0))

    @pytest.mark.parametrize('axis', [0, 1, 2, None])
    def test_size_zero_with_axis(self, axis, xp):
        x = xp.zeros((3, 0, 4))
        context = (eager_warns(SmallSampleWarning, match='too small', xp=np)
                   if axis in {1, None} else contextlib.nullcontext())
        with context:
            mad = stats.median_abs_deviation(x, axis=axis)
        xp_assert_close(mad, xp.full_like(xp.sum(x, axis=axis), fill_value=xp.nan))

    @pytest.mark.parametrize('nan_policy, expected',
                             [('omit', [np.nan, 1.5, 1.5]),
                              ('propagate', [np.nan, np.nan, 1.5])])
    def test_nan_policy_with_axis(self, nan_policy, expected, xp):
        if nan_policy=='omit' and not is_numpy(xp):
            pytest.skip("nan_policy='omit' with n-d input only supported by NumPy")

        x = xp.asarray([[np.nan, np.nan, np.nan, np.nan, np.nan, np.nan],
                        [1, 5, 3, 6, np.nan, np.nan],
                        [5, 6, 7, 9, 9, 10]])

        context = (eager_warns(SmallSampleWarning, match="too small", xp=np)
                   if nan_policy == 'omit' else contextlib.nullcontext())
        with context:
            mad = stats.median_abs_deviation(x, nan_policy=nan_policy, axis=1)
            xp_assert_close(mad, xp.asarray(expected))

    @pytest.mark.parametrize('axis, expected',
                             [(1, [2.5, 2.0, 12.0]), (None, 4.5)])
    def test_center_mean_with_nan(self, axis, expected):
        # nan_policy='omit' with multidimensional input only supported by NumPy
        x = np.array([[1, 2, 4, 9, np.nan],
                      [0, 1, 1, 1, 12],
                      [-10, -10, -10, 20, 20]])
        mad = stats.median_abs_deviation(x, center=np.mean, nan_policy='omit',
                                         axis=axis)
        xp_assert_close(mad, expected, rtol=1e-15, atol=1e-15)

    def test_center_not_callable(self, xp):
        with pytest.raises(TypeError, match='callable'):
            stats.median_abs_deviation(xp.asarray([1, 2, 3, 5]), center=99)


def _check_warnings(warn_list, expected_type, expected_len):
    """
    Checks that all of the warnings from a list returned by
    `warnings.catch_all(record=True)` are of the required type and that the list
    contains expected number of warnings.
    """
    assert_equal(len(warn_list), expected_len, "number of warnings")
    for warn_ in warn_list:
        assert_(warn_.category is expected_type)


@make_xp_test_case(stats.iqr)
class TestIQR:

    @pytest.mark.parametrize('dtype', [None, 'float32', 'float64'])
    def test_basic(self, dtype, xp):
        x = np.arange(8) * 0.5
        np.random.shuffle(x)
        dtype = xp_default_dtype(xp) if dtype is None else getattr(xp, dtype)
        xp_assert_equal(stats.iqr(xp.asarray(x, dtype=dtype)),
                        xp.asarray(1.75, dtype=dtype))

    def test_api(self, xp):
        d = xp.ones((5, 5))
        stats.iqr(d)
        stats.iqr(d, None)
        stats.iqr(d, 1)
        stats.iqr(d, (0, 1))
        stats.iqr(d, None, (10, 90))
        stats.iqr(d, None, (30, 20), 1.0)
        stats.iqr(d, None, (25, 75), 1.5, 'propagate')

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy -> no nan_policy')
    def test_api_eager(self, xp):
        d = xp.ones((5, 5))
        stats.iqr(d, None, (50, 50), 'normal', 'raise', 'linear')
        stats.iqr(d, None, (25, 75), -0.4, 'omit', 'lower', True)

    @pytest.mark.parametrize('x', [[], np.arange(0)])
    def test_empty(self, x, xp):
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            xp_assert_equal(stats.iqr(xp.asarray(x)), xp.asarray(xp.nan))

    def test_constant(self, xp):
        # Constant array always gives 0
        x = xp.ones((7, 4))
        zero = xp.asarray(0.0)
        xp_assert_equal(stats.iqr(x), zero)
        xp_assert_equal(stats.iqr(x, axis=0), xp.zeros(4))
        xp_assert_equal(stats.iqr(x, axis=1), xp.zeros(7))
        xp_assert_equal(stats.iqr(x, interpolation='linear'), zero)
        xp_assert_equal(stats.iqr(x, interpolation='midpoint'), zero)
        xp_assert_equal(stats.iqr(x, interpolation='nearest'), zero)
        xp_assert_equal(stats.iqr(x, interpolation='lower'), zero)
        xp_assert_equal(stats.iqr(x, interpolation='higher'), zero)

        # 0 only along constant dimensions
        # This also tests much of `axis`
        y = xp.ones((4, 5, 6)) * xp.arange(6.)
        xp_assert_equal(stats.iqr(y, axis=0), xp.zeros((5, 6)))
        xp_assert_equal(stats.iqr(y, axis=1), xp.zeros((4, 6)))
        xp_assert_equal(stats.iqr(y, axis=2), xp.full((4, 5), 2.5))
        xp_assert_equal(stats.iqr(y, axis=(0, 1)), xp.zeros(6))
        xp_assert_equal(stats.iqr(y, axis=(0, 2)), xp.full(5, 3.))
        xp_assert_equal(stats.iqr(y, axis=(1, 2)), xp.full(4, 3.))

    def test_scalarlike(self, xp):
        x = xp.arange(1.) + 7.0
        xp_assert_equal(stats.iqr(x[0]), xp.asarray(0.0))
        xp_assert_equal(stats.iqr(x), xp.asarray(0.0))
        xp_assert_equal(stats.iqr(x, keepdims=True), xp.asarray([0.0]))

    def test_2D(self, xp):
        x = xp.reshape(xp.arange(15), (3, 5))
        xp_assert_equal(stats.iqr(x), xp.asarray(7.0))
        xp_assert_equal(stats.iqr(x, axis=0), xp.full(5, 5.))
        xp_assert_equal(stats.iqr(x, axis=1), xp.full(3, 2.))
        xp_assert_equal(stats.iqr(x, axis=(0, 1)), xp.asarray(7.0))
        xp_assert_equal(stats.iqr(x, axis=(1, 0)), xp.asarray(7.0))

    def test_axis(self, xp):
        # The `axis` keyword is also put through its paces in `test_keepdims`.
        o = np.random.normal(size=(71, 23))
        x = np.dstack([o] * 10)                 # x.shape = (71, 23, 10)
        o, x = xp.asarray(o), xp.asarray(x)
        q = xp.broadcast_to(stats.iqr(o), (10,))

        xp_assert_equal(stats.iqr(x, axis=(0, 1)), q)
        x = xp.moveaxis(x, -1, 0)               # x.shape = (10, 71, 23)
        xp_assert_equal(stats.iqr(x, axis=(2, 1)), q)
        x = xp_swapaxes(x,0, 1, xp=xp)                    # x.shape = (71, 10, 23)
        xp_assert_equal(stats.iqr(x, axis=(0, 2)), q)
        x = xp_swapaxes(x,0, 1, xp=xp)                    # x.shape = (10, 71, 23)

        xp_assert_equal(stats.iqr(x, axis=(0, 1, 2)),
                        stats.iqr(x, axis=None))
        xp_assert_equal(stats.iqr(x, axis=(0,)),
                        stats.iqr(x, axis=0))

        d = np.arange(3 * 5 * 7 * 11)
        # Older versions of numpy only shuffle along axis=0.
        # Not sure about newer, don't care.
        np.random.shuffle(d)
        d = d.reshape((3, 5, 7, 11))
        d = xp.asarray(d)
        xp_assert_equal(stats.iqr(d, axis=(0, 1, 2))[0],
                        stats.iqr(xp_ravel(d[:,:,:, 0])))
        xp_assert_equal(stats.iqr(d, axis=(0, 1, 3))[1],
                        stats.iqr(xp_ravel(d[:,:, 1,:])))
        xp_assert_equal(stats.iqr(d, axis=(3, 1, -4))[2],
                        stats.iqr(xp_ravel(d[:,:, 2,:])))
        xp_assert_equal(stats.iqr(d, axis=(3, 1, 2))[2],
                        stats.iqr(xp_ravel(d[2,:,:,:])))
        xp_assert_equal(stats.iqr(d, axis=(3, 2))[2, 1],
                        stats.iqr(xp_ravel(d[2, 1,:,:])))
        xp_assert_equal(stats.iqr(d, axis=(1, -2))[2, 1],
                        stats.iqr(xp_ravel(d[2, :, :, 1])))
        xp_assert_equal(stats.iqr(d, axis=(1, 3))[2, 2],
                        stats.iqr(xp_ravel(d[2, :, 2,:])))

        with pytest.raises(AxisError, match='`axis` is out of bounds...'):
            stats.iqr(d, axis=4)
        with pytest.raises(ValueError, match='`axis` must contain only...'):
            stats.iqr(d, axis=(0, 0))

    def test_rng(self, xp):
        x = xp.arange(5)
        xp_assert_equal(stats.iqr(x), xp.asarray(2.))
        xp_assert_equal(stats.iqr(x, rng=(25, 87.5)), xp.asarray(2.5))
        xp_assert_equal(stats.iqr(x, rng=(12.5, 75)), xp.asarray(2.5))
        xp_assert_equal(stats.iqr(x, rng=(10, 50)), xp.asarray(1.6))  # 3-1.4

        message = r"Elements of `rng` must be in the range \[0, 100\]."
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, rng=(0, 101))

        message = "`rng` must not contain NaNs."
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, rng=(np.nan, 25))

        message = "`rng` must be a two element sequence."
        with pytest.raises(TypeError, match=message):
            stats.iqr(x, rng=(0, 50, 60))

    def test_interpolation(self, xp):
        x = xp.arange(5)
        y = xp.arange(4)
        # Default
        xp_assert_equal(stats.iqr(x), xp.asarray(2.))
        xp_assert_equal(stats.iqr(y), xp.asarray(1.5))
        # Linear
        xp_assert_equal(stats.iqr(x, interpolation='linear'), xp.asarray(2.))
        xp_assert_equal(stats.iqr(y, interpolation='linear'), xp.asarray(1.5))
        # Higher
        xp_assert_equal(stats.iqr(x, interpolation='higher'), xp.asarray(2.))
        xp_assert_equal(stats.iqr(x, rng=(25, 80), interpolation='higher'),
                        xp.asarray(3.))
        xp_assert_equal(stats.iqr(y, interpolation='higher'), xp.asarray(2.))
        # Lower (will generally, but not always be the same as higher)
        xp_assert_equal(stats.iqr(x, interpolation='lower'), xp.asarray(2.))
        xp_assert_equal(stats.iqr(x, rng=(25, 80), interpolation='lower'),
                        xp.asarray(2.))
        xp_assert_equal(stats.iqr(y, interpolation='lower'), xp.asarray(2.))
        # Nearest
        xp_assert_equal(stats.iqr(x, interpolation='nearest'), xp.asarray(2.))
        xp_assert_equal(stats.iqr(y, interpolation='nearest'), xp.asarray(1.))
        # Midpoint
        xp_assert_equal(stats.iqr(x, interpolation='midpoint'), xp.asarray(2.))
        xp_assert_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'),
                        xp.asarray(2.5))
        xp_assert_equal(stats.iqr(y, interpolation='midpoint'), xp.asarray(2.))

        # Check all method= values new in numpy 1.22.0 are accepted
        for method in ('inverted_cdf', 'averaged_inverted_cdf',
                       'closest_observation', 'interpolated_inverted_cdf',
                       'hazen', 'weibull', 'median_unbiased',
                       'normal_unbiased'):
            stats.iqr(y, interpolation=method)

        with pytest.raises(ValueError, match='`method` must be one of...'):
            stats.iqr(x, interpolation='foobar')

    def test_keepdims(self, xp):
        # Also tests most of `axis`
        x = xp.ones((3, 5, 7, 11))
        assert_equal(stats.iqr(x, axis=None, keepdims=False).shape, ())
        assert_equal(stats.iqr(x, axis=2, keepdims=False).shape, (3, 5, 11))
        assert_equal(stats.iqr(x, axis=(0, 1), keepdims=False).shape, (7, 11))
        assert_equal(stats.iqr(x, axis=(0, 3), keepdims=False).shape, (5, 7))
        assert_equal(stats.iqr(x, axis=(1,), keepdims=False).shape, (3, 7, 11))
        assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=False).shape, ())
        assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=False).shape, (7,))

        assert_equal(stats.iqr(x, axis=None, keepdims=True).shape, (1, 1, 1, 1))
        assert_equal(stats.iqr(x, axis=2, keepdims=True).shape, (3, 5, 1, 11))
        assert_equal(stats.iqr(x, axis=(0, 1), keepdims=True).shape, (1, 1, 7, 11))
        assert_equal(stats.iqr(x, axis=(0, 3), keepdims=True).shape, (1, 5, 7, 1))
        assert_equal(stats.iqr(x, axis=(1,), keepdims=True).shape, (3, 1, 7, 11))
        assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=True).shape, (1, 1, 1, 1))
        assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=True).shape, (1, 1, 7, 1))

    def test_nanpolicy(self, xp):
        x = xp.reshape(xp.arange(15.0), (3, 5))

        # No NaNs
        xp_assert_equal(stats.iqr(x, nan_policy='propagate'), xp.asarray(7.))
        xp_assert_equal(stats.iqr(x, nan_policy='omit'), xp.asarray(7.))
        xp_assert_equal(stats.iqr(x, nan_policy='raise'), xp.asarray(7.))

        # Yes NaNs
        x = xpx.at(x)[1, 2].set(xp.nan)
        xp_assert_equal(stats.iqr(x, nan_policy='propagate'),
                        xp.asarray(xp.nan))
        xp_assert_equal(stats.iqr(x, axis=0, nan_policy='propagate'),
                        xp.asarray([5, 5, xp.nan, 5, 5]))
        xp_assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'),
                        xp.asarray([2, xp.nan, 2]))

        xp_assert_equal(stats.iqr(x, nan_policy='omit'), xp.asarray(7.5))

        message = "The input contains nan values"
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, nan_policy='raise')

        # Bad policy
        message = "nan_policy must be one of..."
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, nan_policy='barfood')

    @pytest.mark.skip_xp_backends(np_only=True,
        reason="nan_policy w/ multidimensional arrays only available w/ NumPy")
    def test_nanpolicy_nd(self, xp):
        x = xp.reshape(xp.arange(15.0), (3, 5))
        x[1, 2] = xp.nan
        xp_assert_equal(stats.iqr(x, axis=0, nan_policy='omit'),
                        xp.full(5, 5.))
        xp_assert_equal(stats.iqr(x, axis=1, nan_policy='omit'),
                        xp.asarray([2, 2.5, 2]))

        message = "The input contains nan values"
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, axis=0, nan_policy='raise')
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, axis=1, nan_policy='raise')

    def test_scale(self, xp):
        x = xp.reshape(xp.arange(15.0), (3, 5))

        # No NaNs
        xp_assert_equal(stats.iqr(x, scale=1.0), xp.asarray(7.))
        xp_assert_close(stats.iqr(x, scale='normal'), xp.asarray(7 / 1.3489795))
        xp_assert_equal(stats.iqr(x, scale=2.0), xp.asarray(3.5))

        # Yes NaNs
        x = xpx.at(x)[1, 2].set(xp.nan)
        nan = xp.asarray(xp.nan)
        xp_assert_equal(stats.iqr(x, scale=1.0, nan_policy='propagate'), nan)
        xp_assert_equal(stats.iqr(x, scale='normal', nan_policy='propagate'), nan)
        xp_assert_equal(stats.iqr(x, scale=2.0, nan_policy='propagate'), nan)

        xp_assert_equal(stats.iqr(x, scale=1.0, nan_policy='omit'), xp.asarray(7.5))
        xp_assert_close(stats.iqr(x, scale='normal', nan_policy='omit'),
                        xp.asarray(7.5 / 1.3489795))
        xp_assert_equal(stats.iqr(x, scale=2.0, nan_policy='omit'), xp.asarray(3.75))

        # # Bad scale
        message = "foobar not a valid scale for `iqr`"
        with pytest.raises(ValueError, match=message):
            stats.iqr(x, scale='foobar')

    @pytest.mark.skip_xp_backends(np_only=True,
        reason="nan_policy w/ multidimensional arrays only available w/ NumPy")
    def test_scale_nanpolicy_nd(self, xp):
        # axis=1 chosen to show behavior with both nans and without
        x = xp.reshape(xp.arange(15.0), (3, 5))
        x = xpx.at(x)[1, 2].set(xp.nan)

        xp_assert_equal(stats.iqr(x, axis=1, scale=1.0, nan_policy='propagate'),
                        xp.asarray([2, np.nan, 2]))
        xp_assert_close(stats.iqr(x, axis=1, scale='normal', nan_policy='propagate'),
                        xp.asarray([2, np.nan, 2]) / 1.3489795)
        xp_assert_equal(stats.iqr(x, axis=1, scale=2.0, nan_policy='propagate'),
                        xp.asarray([1, np.nan, 1]))

    def test_rng_order(self, xp):
        # test that order of `rng` doesn't matter (as documented)
        x = xp.arange(8.) * 0.5
        res = stats.iqr(x, rng=(75, 25))
        ref = stats.iqr(x)
        xp_assert_equal(res, ref)


@make_xp_test_case(stats.moment)
class TestMoments:
    """
        Comparison numbers are found using R v.1.5.1
        note that length(testcase) = 4
        testmathworks comes from documentation for the
        Statistics Toolbox for Matlab and can be found at both
        https://www.mathworks.com/help/stats/kurtosis.html
        https://www.mathworks.com/help/stats/skewness.html
        Note that both test cases came from here.
    """
    testcase = [1., 2., 3., 4.]
    scalar_testcase = 4.
    rng = np.random.default_rng(2285049930)
    testcase_moment_accuracy = rng.random(42)

    @pytest.mark.parametrize('size', [10, (10, 2)])
    @pytest.mark.parametrize('m, c', product((0, 1, 2, 3), (None, 0, 1)))
    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered in divide:RuntimeWarning:dask"
    )
    def test_moment_center_scalar_moment(self, size, m, c, xp):
        rng = np.random.default_rng(6581432544381372042)
        x = xp.asarray(rng.random(size=size))
        res = stats.moment(x, m, center=c)
        c = xp.mean(x, axis=0) if c is None else c
        ref = xp.sum((x - c)**m, axis=0)/x.shape[0]
        xp_assert_close(res, ref, atol=1e-16)

    @pytest.mark.parametrize('size', [10, (10, 2)])
    @pytest.mark.parametrize('c', (None, 0, 1))
    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered in divide:RuntimeWarning:dask"
    )
    def test_moment_center_array_moment(self, size, c, xp):
        rng = np.random.default_rng(1706828300224046506)
        x = xp.asarray(rng.random(size=size))
        m = [0, 1, 2, 3]
        res = stats.moment(x, m, center=c)
        ref = xp.concat([stats.moment(x, i, center=c)[xp.newaxis, ...] for i in m])
        xp_assert_equal(res, ref)

    def test_moment(self, xp):
        # mean((testcase-mean(testcase))**power,axis=0),axis=0))**power))
        testcase = xp.asarray(self.testcase)

        y = stats.moment(xp.asarray(self.scalar_testcase))
        xp_assert_close(y, xp.asarray(0.0))

        y = stats.moment(testcase, 0)
        xp_assert_close(y, xp.asarray(1.0))
        y = stats.moment(testcase, 1)
        xp_assert_close(y, xp.asarray(0.0))
        y = stats.moment(testcase, 2)
        xp_assert_close(y, xp.asarray(1.25))
        y = stats.moment(testcase, 3)
        xp_assert_close(y, xp.asarray(0.0))
        y = stats.moment(testcase, 4)
        xp_assert_close(y, xp.asarray(2.5625))

        # check array_like input for moment
        y = stats.moment(testcase, [1, 2, 3, 4])
        xp_assert_close(y, xp.asarray([0, 1.25, 0, 2.5625]))

        # check moment input consists only of integers
        y = stats.moment(testcase, 0.0)
        xp_assert_close(y, xp.asarray(1.0))
        message = 'All elements of `order` must be integral.'
        with pytest.raises(ValueError, match=message):
            stats.moment(testcase, 1.2)
        y = stats.moment(testcase, [1.0, 2, 3, 4.0])
        xp_assert_close(y, xp.asarray([0, 1.25, 0, 2.5625]))

        def test_cases():
            y = stats.moment(xp.asarray([]))
            xp_assert_equal(y, xp.asarray(xp.nan))
            y = stats.moment(xp.asarray([], dtype=xp.float32))
            xp_assert_equal(y, xp.asarray(xp.nan, dtype=xp.float32))
            y = stats.moment(xp.zeros((1, 0)), axis=0)
            xp_assert_equal(y, xp.empty((0,)))
            y = stats.moment(xp.asarray([[]]), axis=1)
            xp_assert_equal(y, xp.asarray([xp.nan]))
            y = stats.moment(xp.asarray([[]]), order=[0, 1], axis=0)
            xp_assert_equal(y, xp.empty((2, 0)))

    def test_nan_policy(self):
        x = np.arange(10.)
        x[9] = np.nan
        assert_equal(stats.moment(x, 2), np.nan)
        assert_almost_equal(stats.moment(x, nan_policy='omit'), 0.0)
        assert_raises(ValueError, stats.moment, x, nan_policy='raise')
        assert_raises(ValueError, stats.moment, x, nan_policy='foobar')

    @pytest.mark.parametrize('dtype', ['float32', 'float64', 'complex128'])
    @pytest.mark.parametrize('expect, order', [(0, 1), (1, 0)])
    def test_constant_moments(self, dtype, expect, order, xp):
        if dtype=='complex128' and is_torch(xp):
            pytest.skip()
        dtype = getattr(xp, dtype)
        rng = np.random.default_rng(3824693518)
        x = xp.asarray(rng.random(5), dtype=dtype)
        y = stats.moment(x, order=order)
        xp_assert_equal(y, xp.asarray(expect, dtype=dtype))

        y = stats.moment(xp.broadcast_to(x, (6, 5)), axis=0, order=order)
        xp_assert_equal(y, xp.full((5,), expect, dtype=dtype))

        y = stats.moment(xp.broadcast_to(x, (1, 2, 3, 4, 5)), axis=2,
                         order=order)
        xp_assert_equal(y, xp.full((1, 2, 4, 5), expect, dtype=dtype))

        y = stats.moment(xp.broadcast_to(x, (1, 2, 3, 4, 5)), axis=None,
                         order=order)
        xp_assert_equal(y, xp.full((), expect, dtype=dtype))

    def test_moment_propagate_nan(self, xp):
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = xp.reshape(xp.arange(8.), (2, -1))
        a = xpx.at(a)[1, 0].set(xp.nan)

        mm = stats.moment(xp.asarray(a), 2, axis=1)
        xp_assert_close(mm, xp.asarray([1.25, xp.nan]), atol=1e-15)

    def test_moment_empty_order(self, xp):
        # tests moment with empty `order` list
        with pytest.raises(ValueError, match=r"`order` must be a scalar or a"
                                             r" non-empty 1D array."):
            stats.moment(xp.asarray([1, 2, 3, 4]), order=[])

    def test_rename_moment_order(self, xp):
        # Parameter 'order' was formerly known as 'moment'. The old name
        # has not been deprecated, so it must continue to work.
        x = xp.arange(10)
        res = stats.moment(x, moment=3)
        ref = stats.moment(x, order=3)
        xp_assert_equal(res, ref)

    def test_moment_accuracy(self):
        # 'moment' must have a small enough error compared to the slower
        #  but very accurate numpy.power() implementation.
        tc_no_mean = (self.testcase_moment_accuracy
                      - np.mean(self.testcase_moment_accuracy))
        assert_allclose(np.power(tc_no_mean, 42).mean(),
                        stats.moment(self.testcase_moment_accuracy, 42))

    @pytest.mark.parametrize('order', [0, 1, 2, 3])
    @pytest.mark.parametrize('axis', [-1, 0, 1])
    @pytest.mark.parametrize('center', [None, 0])
    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered in divide:RuntimeWarning:dask"
    )
    def test_moment_array_api(self, xp, order, axis, center):
        rng = np.random.default_rng(34823589259425)
        x = rng.random(size=(5, 6, 7))
        res = stats.moment(xp.asarray(x), order, axis=axis, center=center)
        ref = xp.asarray(_moment(x, order, axis, mean=center))
        xp_assert_close(res, ref)


class SkewKurtosisTest:
    scalar_testcase = 4.
    testcase = [1., 2., 3., 4.]
    testmathworks = [1.165, 0.6268, 0.0751, 0.3516, -0.6965]

    def test_empty_1d(self, xp):
        x = xp.asarray([])
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            res = self.stat_fun(x)

        xp_assert_equal(res, xp.asarray(xp.nan))


@make_xp_test_case(stats.skew)
class TestSkew(SkewKurtosisTest):
    def stat_fun(self, x):
        return stats.skew(x)

    @pytest.mark.filterwarnings(
        "ignore:invalid value encountered in scalar divide:RuntimeWarning:dask"
    )
    def test_skewness(self, xp):
        # Scalar test case
        y = stats.skew(xp.asarray(self.scalar_testcase))
        xp_assert_close(y, xp.asarray(xp.nan))
        # sum((testmathworks-mean(testmathworks,axis=0))**3,axis=0) /
        #     ((sqrt(var(testmathworks)*4/5))**3)/5
        y = stats.skew(xp.asarray(self.testmathworks))
        xp_assert_close(y, xp.asarray(-0.29322304336607), atol=1e-10)
        y = stats.skew(xp.asarray(self.testmathworks), bias=0)
        xp_assert_close(y, xp.asarray(-0.437111105023940), atol=1e-10)
        y = stats.skew(xp.asarray(self.testcase))
        xp_assert_close(y, xp.asarray(0.0), atol=1e-10)

    def test_nan_policy(self):
        # initially, nan_policy is ignored with alternative backends
        x = np.arange(10.)
        x[9] = np.nan
        with np.errstate(invalid='ignore'):
            assert_equal(stats.skew(x), np.nan)
        assert_equal(stats.skew(x, nan_policy='omit'), 0.)
        assert_raises(ValueError, stats.skew, x, nan_policy='raise')
        assert_raises(ValueError, stats.skew, x, nan_policy='foobar')

    def test_skewness_scalar(self):
        # `skew` must return a scalar for 1-dim input (only for NumPy arrays)
        assert_equal(stats.skew(arange(10)), 0.0)

    def test_skew_propagate_nan(self, xp):
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = xp.arange(8.)
        a = xp.reshape(a, (2, -1))
        a = xpx.at(a)[1, 0].set(xp.nan)
        with np.errstate(invalid='ignore'):
            s = stats.skew(xp.asarray(a), axis=1)
        xp_assert_equal(s, xp.asarray([0, xp.nan]))

    def test_skew_constant_value(self, xp):
        # Skewness of a constant input should be NaN (gh-16061)
        a = xp.repeat(xp.asarray([-0.27829495]), 10)

        with eager_warns(RuntimeWarning, match="Precision loss occurred", xp=xp):
            xp_assert_equal(stats.skew(a), xp.asarray(xp.nan))
            xp_assert_equal(stats.skew(a*2.**50), xp.asarray(xp.nan))
            xp_assert_equal(stats.skew(a/2.**50), xp.asarray(xp.nan))
            xp_assert_equal(stats.skew(a, bias=False), xp.asarray(xp.nan))

            # # similarly, from gh-11086:
            a = xp.asarray([14.3]*7)
            xp_assert_equal(stats.skew(a), xp.asarray(xp.nan))
            a = 1. + xp.arange(-3., 4)*1e-16
            xp_assert_equal(stats.skew(a), xp.asarray(xp.nan))

    @skip_xp_backends(eager_only=True)
    def test_precision_loss_gh15554(self, xp):
        # gh-15554 was one of several issues that have reported problems with
        # constant or near-constant input. We can't always fix these, but
        # make sure there's a warning.
        with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
            rng = np.random.default_rng(34095309370)
            a = rng.random(size=(100, 10))
            a[:, 0] = 1.01
            stats.skew(xp.asarray(a))

    @pytest.mark.parametrize('axis', [-1, 0, 2, None])
    @pytest.mark.parametrize('bias', [False, True])
    def test_vectorization(self, xp, axis, bias):
        # Behavior with array input is barely tested above. Compare
        # against naive implementation.
        rng = np.random.default_rng(1283413549926)
        x = xp.asarray(rng.random((3, 4, 5)))

        def skewness(a, axis, bias):
            # Simple implementation of skewness
            if axis is None:
                a = xp.reshape(a, (-1,))
                axis = 0
            mean = xp.mean(a, axis=axis, keepdims=True)
            mu3 = xp.mean((a - mean)**3, axis=axis)
            std = xp.std(a, axis=axis)
            res = mu3 / std ** 3
            if not bias:
                n = a.shape[axis]
                res *= ((n - 1.0) * n) ** 0.5 / (n - 2.0)
            return res

        res = stats.skew(x, axis=axis, bias=bias)
        ref = skewness(x, axis=axis, bias=bias)
        xp_assert_close(res, ref)


@make_xp_test_case(stats.kurtosis)
class TestKurtosis(SkewKurtosisTest):
    def stat_fun(self, x):
        return stats.kurtosis(x)

    @pytest.mark.filterwarnings("ignore:invalid value encountered in scalar divide")
    def test_kurtosis(self, xp):
        # Scalar test case
        y = stats.kurtosis(xp.asarray(self.scalar_testcase))
        assert xp.isnan(y)

        #   sum((testcase-mean(testcase,axis=0))**4,axis=0)
        #   / ((sqrt(var(testcase)*3/4))**4)
        #   / 4
        #
        #   sum((test2-mean(testmathworks,axis=0))**4,axis=0)
        #   / ((sqrt(var(testmathworks)*4/5))**4)
        #   / 5
        #
        #   Set flags for axis = 0 and
        #   fisher=0 (Pearson's defn of kurtosis for compatibility with Matlab)
        y = stats.kurtosis(xp.asarray(self.testmathworks), 0, fisher=0, bias=1)
        xp_assert_close(y, xp.asarray(2.1658856802973))

        # Note that MATLAB has confusing docs for the following case
        #  kurtosis(x,0) gives an unbiased estimate of Pearson's skewness
        #  kurtosis(x)  gives a biased estimate of Fisher's skewness (Pearson-3)
        #  The MATLAB docs imply that both should give Fisher's
        y = stats.kurtosis(xp.asarray(self.testmathworks), fisher=0, bias=0)
        xp_assert_close(y, xp.asarray(3.663542721189047))
        y = stats.kurtosis(xp.asarray(self.testcase), 0, 0)
        xp_assert_close(y, xp.asarray(1.64))

        x = xp.arange(10.)
        x = xp.where(x == 8, xp.nan, x)
        xp_assert_equal(stats.kurtosis(x), xp.asarray(xp.nan))

    def test_kurtosis_nan_policy(self):
        # nan_policy only for NumPy right now
        x = np.arange(10.)
        x[9] = np.nan
        assert_almost_equal(stats.kurtosis(x, nan_policy='omit'), -1.230000)
        assert_raises(ValueError, stats.kurtosis, x, nan_policy='raise')
        assert_raises(ValueError, stats.kurtosis, x, nan_policy='foobar')

    def test_kurtosis_array_scalar(self):
        # "array scalars" do not exist in other backends
        assert_equal(type(stats.kurtosis([1, 2, 3])), np.float64)

    def test_kurtosis_propagate_nan(self):
        # nan_policy only for NumPy right now
        # Check that the shape of the result is the same for inputs
        # with and without nans, cf gh-5817
        a = np.arange(8).reshape(2, -1).astype(float)
        a[1, 0] = np.nan
        k = stats.kurtosis(a, axis=1, nan_policy="propagate")
        np.testing.assert_allclose(k, [-1.36, np.nan], atol=1e-15)

    def test_kurtosis_constant_value(self, xp):
        # Kurtosis of a constant input should be NaN (gh-16061)
        a = xp.asarray([-0.27829495]*10)
        with eager_warns(RuntimeWarning, match="Precision loss occurred", xp=xp):
            assert xp.isnan(stats.kurtosis(a, fisher=False))
            assert xp.isnan(stats.kurtosis(a * float(2**50), fisher=False))
            assert xp.isnan(stats.kurtosis(a / float(2**50), fisher=False))
            assert xp.isnan(stats.kurtosis(a, fisher=False, bias=False))

    @pytest.mark.parametrize('axis', [-1, 0, 2, None])
    @pytest.mark.parametrize('bias', [False, True])
    @pytest.mark.parametrize('fisher', [False, True])
    def test_vectorization(self, xp, axis, bias, fisher):
        # Behavior with array input is not tested above. Compare
        # against naive implementation.
        rng = np.random.default_rng(1283413549926)
        x = xp.asarray(rng.random((4, 5, 6)))

        def kurtosis(a, axis, bias, fisher):
            # Simple implementation of kurtosis
            if axis is None:
                a = xp.reshape(a, (-1,))
                axis = 0
            mean = xp.mean(a, axis=axis, keepdims=True)
            mu4 = xp.mean((a - mean)**4, axis=axis)
            mu2 = xp.var(a, axis=axis, correction=0)
            if bias:
                res = mu4 / mu2**2 - 3
            else:
                n = a.shape[axis]
                # https://en.wikipedia.org/wiki/Kurtosis#Standard_unbiased_estimator
                res = (n-1) / ((n-2) * (n-3)) * ((n + 1) * mu4/mu2**2 - 3*(n-1))

            # I know it looks strange to subtract then add 3,
            # but it is simpler than the alternatives
            return res if fisher else res + 3

        res = stats.kurtosis(x, axis=axis, bias=bias, fisher=fisher)
        ref = kurtosis(x, axis=axis, bias=bias, fisher=fisher)
        xp_assert_close(res, ref)


@hypothesis.strategies.composite
def ttest_data_axis_strategy(draw):
    # draw an array under shape and value constraints
    elements = dict(allow_nan=False, allow_infinity=False)
    shape = npst.array_shapes(min_dims=1, min_side=2)
    # The test that uses this, `test_pvalue_ci`, uses `float64` to test
    # extreme `alpha`. It could be adjusted to test a dtype-dependent
    # range of `alpha` if this strategy is needed to generate other floats.
    data = draw(npst.arrays(dtype=np.float64, elements=elements, shape=shape))

    # determine axes over which nonzero variance can be computed accurately
    ok_axes = []
    # Locally, I don't need catch_warnings or simplefilter, and I can just
    # suppress RuntimeWarning. I include all that in hope of getting the same
    # behavior on CI.
    with warnings.catch_warnings():
        warnings.simplefilter("error")
        for axis in range(len(data.shape)):
            with contextlib.suppress(Exception):
                var = stats.moment(data, order=2, axis=axis)
                if np.all(var > 0) and np.all(np.isfinite(var)):
                    ok_axes.append(axis)
    # if there are no valid axes, tell hypothesis to try a different example
    hypothesis.assume(ok_axes)

    # draw one of the valid axes
    axis = draw(hypothesis.strategies.sampled_from(ok_axes))

    return data, axis


@make_xp_test_case(stats.ttest_1samp)
class TestStudentTest:
    # Preserving original test cases.
    # Recomputed statistics and p-values with R t.test, e.g.
    # options(digits=16)
    # t.test(c(-1., 0., 1.), mu=2)
    X1 = [-1., 0., 1.]
    X2 = [0., 1., 2.]
    T1_0 = 0.
    P1_0 = 1.
    T1_1 = -1.7320508075689
    P1_1 = 0.2254033307585
    T1_2 = -3.4641016151378
    P1_2 = 0.07417990022745
    T2_0 = 1.7320508075689
    P2_0 = 0.2254033307585
    P1_1_l = P1_1 / 2
    P1_1_g = 1 - (P1_1 / 2)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_onesample(self, xp):
        with warnings.catch_warnings(), \
                np.errstate(invalid="ignore", divide="ignore"):
            warnings.filterwarnings(
                "ignore", "Degrees of freedom <= 0 for slice", RuntimeWarning)
            a = xp.asarray(4.) if not is_numpy(xp) else 4.
            t, p = stats.ttest_1samp(a, 3.)
        xp_assert_equal(t, xp.asarray(xp.nan))
        xp_assert_equal(p, xp.asarray(xp.nan))

        t, p = stats.ttest_1samp(xp.asarray(self.X1), 0.)
        xp_assert_close(t, xp.asarray(self.T1_0))
        xp_assert_close(p, xp.asarray(self.P1_0))

        res = stats.ttest_1samp(xp.asarray(self.X1), 0.)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes, xp=xp)

        t, p = stats.ttest_1samp(xp.asarray(self.X2), 0.)

        xp_assert_close(t, xp.asarray(self.T2_0))
        xp_assert_close(p, xp.asarray(self.P2_0))

        t, p = stats.ttest_1samp(xp.asarray(self.X1), 1.)

        xp_assert_close(t, xp.asarray(self.T1_1))
        xp_assert_close(p, xp.asarray(self.P1_1))

        t, p = stats.ttest_1samp(xp.asarray(self.X1), 2.)

        xp_assert_close(t, xp.asarray(self.T1_2))
        xp_assert_close(p, xp.asarray(self.P1_2))

    def test_onesample_nan_policy_propagate(self, xp):
        x = stats.norm.rvs(loc=5, scale=10, size=51, random_state=7654567)
        x[50] = np.nan
        x = xp.asarray(x, dtype=xp_default_dtype(xp))

        res = stats.ttest_1samp(x, 5.0)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

    @skip_xp_backends(eager_only=True, reason="lazy arrays don't do 'raise'.")
    def test_onesample_nan_policy_omit_raise(self, xp):
        x = stats.norm.rvs(loc=5, scale=10, size=51, random_state=7654567)
        x[50] = np.nan
        x = xp.asarray(x, dtype=xp_default_dtype(xp))

        res = stats.ttest_1samp(x, 5.0, nan_policy='omit')
        xp_assert_close(res.statistic, xp.asarray(-1.6412624074367159))
        xp_assert_close(res.pvalue, xp.asarray(0.107147027334048005))

        with pytest.raises(ValueError, match="The input contains nan values"):
            stats.ttest_1samp(x, 5.0, nan_policy='raise')

        with pytest.raises(ValueError, match="nan_policy must be one of"):
            stats.ttest_1samp(x, 5.0, nan_policy='foobar')


    @pytest.mark.filterwarnings("ignore:divide by zero encountered in divide")
    def test_1samp_alternative(self, xp):
        message = "`alternative` must be 'less', 'greater', or 'two-sided'."
        with pytest.raises(ValueError, match=message):
            stats.ttest_1samp(xp.asarray(self.X1), 0., alternative="error")

        t, p = stats.ttest_1samp(xp.asarray(self.X1), 1., alternative="less")
        xp_assert_close(p, xp.asarray(self.P1_1_l))
        xp_assert_close(t, xp.asarray(self.T1_1))

        t, p = stats.ttest_1samp(xp.asarray(self.X1), 1., alternative="greater")
        xp_assert_close(p, xp.asarray(self.P1_1_g))
        xp_assert_close(t, xp.asarray(self.T1_1))

    @skip_xp_backends('jax.numpy', reason='Generic stdtrit mutates array.')
    @pytest.mark.parametrize("alternative", ['two-sided', 'less', 'greater'])
    def test_1samp_ci_1d(self, xp, alternative):
        # test confidence interval method against reference values
        rng = np.random.default_rng(8066178009154342972)
        n = 10
        x = rng.normal(size=n, loc=1.5, scale=2)
        popmean = rng.normal()  # this shouldn't affect confidence interval
        # Reference values generated with R t.test:
        # options(digits=16)
        # x = c(2.75532884,  0.93892217,  0.94835861,  1.49489446, -0.62396595,
        #      -1.88019867, -1.55684465,  4.88777104,  5.15310979,  4.34656348)
        # t.test(x, conf.level=0.85, alternative='l')
        dtype = xp.asarray(1.0).dtype
        x = xp.asarray(x, dtype=dtype)
        popmean = xp.asarray(popmean, dtype=dtype)

        ref = {'two-sided': [0.3594423211709136, 2.9333455028290860],
               'greater': [0.7470806207371626, np.inf],
               'less': [-np.inf, 2.545707203262837]}
        res = stats.ttest_1samp(x, popmean=popmean, alternative=alternative)
        ci = res.confidence_interval(confidence_level=0.85)
        xp_assert_close(ci.low, xp.asarray(ref[alternative][0]))
        xp_assert_close(ci.high, xp.asarray(ref[alternative][1]))
        xp_assert_equal(res.df, xp.asarray(n-1))

    def test_1samp_ci_iv(self, xp):
        # test `confidence_interval` method input validation
        res = stats.ttest_1samp(xp.arange(10.), 0.)
        message = '`confidence_level` must be a number between 0 and 1.'
        with pytest.raises(ValueError, match=message):
            res.confidence_interval(confidence_level=10)

    @skip_xp_backends(np_only=True, reason='Too slow.')
    @pytest.mark.xslow
    @hypothesis.given(alpha=hypothesis.strategies.floats(1e-15, 1-1e-15),
                      data_axis=ttest_data_axis_strategy())
    @pytest.mark.parametrize('alternative', ['less', 'greater'])
    def test_pvalue_ci(self, alpha, data_axis, alternative, xp):
        # test relationship between one-sided p-values and confidence intervals
        data, axis = data_axis
        data = xp.asarray(data)
        res = stats.ttest_1samp(data, 0.,
                                alternative=alternative, axis=axis)
        l, u = res.confidence_interval(confidence_level=alpha)
        popmean = l if alternative == 'greater' else u
        popmean = xp.expand_dims(popmean, axis=axis)
        res = stats.ttest_1samp(data, popmean, alternative=alternative, axis=axis)
        shape = list(data.shape)
        shape.pop(axis)
        # `float64` is used to correspond with extreme range of `alpha`
        ref = xp.broadcast_to(xp.asarray(1-alpha, dtype=xp.float64), shape)
        xp_assert_close(res.pvalue, ref)


class TestPercentileOfScore:

    def f(self, *args, **kwargs):
        return stats.percentileofscore(*args, **kwargs)

    @pytest.mark.parametrize("kind, result", [("rank", 40),
                                              ("mean", 35),
                                              ("strict", 30),
                                              ("weak", 40)])
    def test_unique(self, kind, result):
        a = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
        assert_equal(self.f(a, 4, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", 45),
                                              ("mean", 40),
                                              ("strict", 30),
                                              ("weak", 50)])
    def test_multiple2(self, kind, result):
        a = [1, 2, 3, 4, 4, 5, 6, 7, 8, 9]
        assert_equal(self.f(a, 4, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", 50),
                                              ("mean", 45),
                                              ("strict", 30),
                                              ("weak", 60)])
    def test_multiple3(self, kind, result):
        a = [1, 2, 3, 4, 4, 4, 5, 6, 7, 8]
        assert_equal(self.f(a, 4, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", 30),
                                              ("mean", 30),
                                              ("strict", 30),
                                              ("weak", 30)])
    def test_missing(self, kind, result):
        a = [1, 2, 3, 5, 6, 7, 8, 9, 10, 11]
        assert_equal(self.f(a, 4, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", 40),
                                              ("mean", 35),
                                              ("strict", 30),
                                              ("weak", 40)])
    def test_large_numbers(self, kind, result):
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
        assert_equal(self.f(a, 40, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", 50),
                                              ("mean", 45),
                                              ("strict", 30),
                                              ("weak", 60)])
    def test_large_numbers_multiple3(self, kind, result):
        a = [10, 20, 30, 40, 40, 40, 50, 60, 70, 80]
        assert_equal(self.f(a, 40, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", 30),
                                              ("mean", 30),
                                              ("strict", 30),
                                              ("weak", 30)])
    def test_large_numbers_missing(self, kind, result):
        a = [10, 20, 30, 50, 60, 70, 80, 90, 100, 110]
        assert_equal(self.f(a, 40, kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", [0, 10, 100, 100]),
                                              ("mean", [0, 5, 95, 100]),
                                              ("strict", [0, 0, 90, 100]),
                                              ("weak", [0, 10, 100, 100])])
    def test_boundaries(self, kind, result):
        a = [10, 20, 30, 50, 60, 70, 80, 90, 100, 110]
        assert_equal(self.f(a, [0, 10, 110, 200], kind=kind), result)

    @pytest.mark.parametrize("kind, result", [("rank", [0, 10, 100]),
                                              ("mean", [0, 5, 95]),
                                              ("strict", [0, 0, 90]),
                                              ("weak", [0, 10, 100])])
    def test_inf(self, kind, result):
        a = [1, 2, 3, 4, 5, 6, 7, 8, 9, +np.inf]
        assert_equal(self.f(a, [-np.inf, 1, +np.inf], kind=kind), result)

    cases = [("propagate", [], 1, np.nan),
             ("propagate", [np.nan], 1, np.nan),
             ("propagate", [np.nan], [0, 1, 2], [np.nan, np.nan, np.nan]),
             ("propagate", [1, 2], [1, 2, np.nan], [50, 100, np.nan]),
             ("omit", [1, 2, np.nan], [0, 1, 2], [0, 50, 100]),
             ("omit", [1, 2], [0, 1, np.nan], [0, 50, np.nan]),
             ("omit", [np.nan, np.nan], [0, 1, 2], [np.nan, np.nan, np.nan])]

    @pytest.mark.parametrize("policy, a, score, result", cases)
    def test_nans_ok(self, policy, a, score, result):
        assert_equal(self.f(a, score, nan_policy=policy), result)

    cases = [
        ("raise", [1, 2, 3, np.nan], [1, 2, 3],
         "The input contains nan values"),
        ("raise", [1, 2, 3], [1, 2, 3, np.nan],
         "The input contains nan values"),
    ]

    @pytest.mark.parametrize("policy, a, score, message", cases)
    def test_nans_fail(self, policy, a, score, message):
        with assert_raises(ValueError, match=message):
            self.f(a, score, nan_policy=policy)

    @pytest.mark.parametrize("shape", [
        (6, ),
        (2, 3),
        (2, 1, 3),
        (2, 1, 1, 3),
    ])
    def test_nd(self, shape):
        a = np.array([0, 1, 2, 3, 4, 5])
        scores = a.reshape(shape)
        results = scores*10
        a = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
        assert_equal(self.f(a, scores, kind="rank"), results)

    def test_multidimensional_error(self):
        # gh-21563 reported that `percentileofscore` accepted multidimensional
        # arrays but did not produce meaningful results.
        message = "`a` must be 1-dimensional."
        with pytest.raises(ValueError, match=message):
            stats.percentileofscore(np.ones((3, 3)), 1)


PowerDivCase = namedtuple('Case',  # type: ignore[name-match]
                          ['f_obs', 'f_exp', 'ddof', 'axis',
                           'chi2',     # Pearson's
                           'log',      # G-test (log-likelihood)
                           'mod_log',  # Modified log-likelihood
                           'cr',       # Cressie-Read (lambda=2/3)
                           ])

# The details of the first two elements in power_div_1d_cases are used
# in a test in TestPowerDivergence.  Check that code before making
# any changes here.
power_div_1d_cases = [
    # Use the default f_exp.
    PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=None, ddof=0, axis=None,
                 chi2=4,
                 log=2*(4*np.log(4/8) + 12*np.log(12/8)),
                 mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
                 cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
    # Give a non-uniform f_exp.
    PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=[2, 16, 12, 2], ddof=0, axis=None,
                 chi2=24,
                 log=2*(4*np.log(4/2) + 8*np.log(8/16) + 8*np.log(8/2)),
                 mod_log=2*(2*np.log(2/4) + 16*np.log(16/8) + 2*np.log(2/8)),
                 cr=(4*((4/2)**(2/3) - 1) + 8*((8/16)**(2/3) - 1) +
                     8*((8/2)**(2/3) - 1))/(5/9)),
    # f_exp is a scalar.
    PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=8, ddof=0, axis=None,
                 chi2=4,
                 log=2*(4*np.log(4/8) + 12*np.log(12/8)),
                 mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
                 cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
    # f_exp equal to f_obs.
    PowerDivCase(f_obs=[3, 5, 7, 9], f_exp=[3, 5, 7, 9], ddof=0, axis=0,
                 chi2=0, log=0, mod_log=0, cr=0),
]


power_div_empty_cases = [
    # Shape is (0,)--a data set with length 0.  The computed
    # test statistic should be 0.
    PowerDivCase(f_obs=[],
                 f_exp=None, ddof=0, axis=0,
                 chi2=0, log=0, mod_log=0, cr=0),
    # Shape is (0, 3).  This is 3 data sets, but each data set has
    # length 0, so the computed test statistic should be [0, 0, 0].
    PowerDivCase(f_obs=np.array([[],[],[]]).T,
                 f_exp=None, ddof=0, axis=0,
                 chi2=[0, 0, 0],
                 log=[0, 0, 0],
                 mod_log=[0, 0, 0],
                 cr=[0, 0, 0]),
    # Shape is (3, 0).  This represents an empty collection of
    # data sets in which each data set has length 3.  The test
    # statistic should be an empty array.
    PowerDivCase(f_obs=np.array([[],[],[]]),
                 f_exp=None, ddof=0, axis=0,
                 chi2=[],
                 log=[],
                 mod_log=[],
                 cr=[]),
]


@make_xp_test_case(stats.power_divergence)
class TestPowerDivergence:

    def check_power_divergence(self, f_obs, f_exp, ddof, axis, lambda_,
                               expected_stat, xp):
        dtype = xp.asarray(1.).dtype

        f_obs = xp.asarray(f_obs, dtype=dtype)
        f_exp = xp.asarray(f_exp, dtype=dtype) if f_exp is not None else f_exp

        if axis is None:
            num_obs = xp_size(f_obs)
        else:
            arrays = (xp.broadcast_arrays(f_obs, f_exp) if f_exp is not None
                      else (f_obs,))
            num_obs = arrays[0].shape[axis]

        with warnings.catch_warnings():
            warnings.filterwarnings("ignore", "Mean of empty slice", RuntimeWarning)
            stat, p = stats.power_divergence(
                                f_obs, f_exp=f_exp, ddof=ddof,
                                axis=axis, lambda_=lambda_)
            xp_assert_close(stat, xp.asarray(expected_stat, dtype=dtype))

            if lambda_ == 1 or lambda_ == "pearson":
                # Also test stats.chisquare.
                stat, p = stats.chisquare(f_obs, f_exp=f_exp, ddof=ddof,
                                          axis=axis)
                xp_assert_close(stat, xp.asarray(expected_stat, dtype=dtype))

        ddof = np.asarray(ddof)
        expected_p = stats.distributions.chi2.sf(expected_stat,
                                                 num_obs - 1 - ddof)
        xp_assert_close(p, xp.asarray(expected_p, dtype=dtype))

    @pytest.mark.parametrize('case', power_div_1d_cases)
    @pytest.mark.parametrize('lambda_stat',
        [(None, 'chi2'), ('pearson', 'chi2'), (1, 'chi2'),
         ('log-likelihood', 'log'), ('mod-log-likelihood', 'mod_log'),
         ('cressie-read', 'cr'), (2/3, 'cr')])
    def test_basic(self, case, lambda_stat, xp):
        lambda_, attr = lambda_stat
        expected_stat = getattr(case, attr)
        self.check_power_divergence(case.f_obs, case.f_exp, case.ddof, case.axis,
                                    lambda_, expected_stat, xp)

    def test_axis(self, xp):
        case0 = power_div_1d_cases[0]
        case1 = power_div_1d_cases[1]
        f_obs = np.vstack((case0.f_obs, case1.f_obs))
        f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
                           case1.f_exp))
        # Check the four computational code paths in power_divergence
        # using a 2D array with axis=1.
        f_obs = xp.asarray(f_obs)
        f_exp = xp.asarray(f_exp) if f_exp is not None else f_exp
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "pearson", [case0.chi2, case1.chi2], xp=xp)
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "log-likelihood", [case0.log, case1.log], xp=xp)
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "mod-log-likelihood", [case0.mod_log, case1.mod_log], xp=xp)
        self.check_power_divergence(
               f_obs, f_exp, 0, 1,
               "cressie-read", [case0.cr, case1.cr], xp=xp)
        # Reshape case0.f_obs to shape (2,2), and use axis=None.
        # The result should be the same.
        f_obs_reshape = xp.reshape(xp.asarray(case0.f_obs), (2, 2))
        self.check_power_divergence(
               f_obs_reshape, None, 0, None,
               "pearson", case0.chi2, xp=xp)

    def test_ddof_broadcasting(self, xp):
        # Test that ddof broadcasts correctly.
        # ddof does not affect the test statistic.  It is broadcast
        # with the computed test statistic for the computation of
        # the p value.

        case0 = power_div_1d_cases[0]
        case1 = power_div_1d_cases[1]
        # Create 4x2 arrays of observed and expected frequencies.
        f_obs = np.vstack((case0.f_obs, case1.f_obs)).T
        f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
                           case1.f_exp)).T

        expected_chi2 = [case0.chi2, case1.chi2]

        dtype = xp.asarray(1.).dtype
        f_obs = xp.asarray(f_obs, dtype=dtype)
        f_exp = xp.asarray(f_exp, dtype=dtype)
        expected_chi2 = xp.asarray(expected_chi2, dtype=dtype)

        # ddof has shape (2, 1).  This is broadcast with the computed
        # statistic, so p will have shape (2,2).
        ddof = xp.asarray([[0], [1]])

        stat, p = stats.power_divergence(f_obs, f_exp, ddof=ddof)
        xp_assert_close(stat, expected_chi2)

        # Compute the p values separately, passing in scalars for ddof.
        _, p0 = stats.power_divergence(f_obs, f_exp, ddof=ddof[0, 0])
        _, p1 = stats.power_divergence(f_obs, f_exp, ddof=ddof[1, 0])

        expected_p = xp.concat((p0[xp.newaxis, :], p1[xp.newaxis, :]), axis=0)
        xp_assert_close(p, expected_p)

    @pytest.mark.parametrize('case', power_div_empty_cases)
    @pytest.mark.parametrize('lambda_stat',
        [('pearson', 'chi2'), ('log-likelihood', 'log'),
         ('mod-log-likelihood', 'mod_log'),
         ('cressie-read', 'cr'), (2/3, 'cr')])
    def test_empty_cases(self, case, lambda_stat, xp):
        lambda_, attr = lambda_stat
        expected_stat = getattr(case, attr)
        with warnings.catch_warnings():
            self.check_power_divergence(
                case.f_obs, case.f_exp, case.ddof, case.axis,
                lambda_, expected_stat, xp)

    def test_power_divergence_result_attributes(self, xp):
        f_obs = power_div_1d_cases[0].f_obs
        f_exp = power_div_1d_cases[0].f_exp
        ddof = power_div_1d_cases[0].ddof
        axis = power_div_1d_cases[0].axis
        dtype = xp.asarray(1.).dtype
        f_obs = xp.asarray(f_obs, dtype=dtype)
        # f_exp is None

        res = stats.power_divergence(f_obs, f_exp=f_exp, ddof=ddof,
                                     axis=axis, lambda_="pearson")
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes, xp=xp)

    def test_power_divergence_gh_12282(self, xp):
        # The sums of observed and expected frequencies must match
        f_obs = xp.asarray([[10., 20.], [30., 20.]])
        f_exp = xp.asarray([[5., 15.], [35., 25.]])
        message = 'For each axis slice...'
        with pytest.raises(ValueError, match=message):
            stats.power_divergence(f_obs, f_exp=xp.asarray([30., 60.]))
        with pytest.raises(ValueError, match=message):
            stats.power_divergence(f_obs, f_exp=f_exp, axis=1)
        stat, pval = stats.power_divergence(f_obs, f_exp=f_exp)
        xp_assert_close(stat, xp.asarray([5.71428571, 2.66666667]))
        xp_assert_close(pval, xp.asarray([0.01682741, 0.10247043]))

    def test_power_divergence_against_cressie_read_data(self, xp):
        # Test stats.power_divergence against tables 4 and 5 from
        # Cressie and Read, "Multimonial Goodness-of-Fit Tests",
        # J. R. Statist. Soc. B (1984), Vol 46, No. 3, pp. 440-464.
        # This tests the calculation for several values of lambda.

        # Table 4 data recalculated for greater precision according to:
        # Shelby J. Haberman, Analysis of Qualitative Data: Volume 1
        # Introductory Topics, Academic Press, New York, USA (1978).
        obs = xp.asarray([15., 11., 14., 17., 5., 11., 10., 4., 8.,
                          10., 7., 9., 11., 3., 6., 1., 1., 4.])
        beta = -0.083769  # Haberman (1978), p. 15
        i = xp.arange(1., obs.shape[0] + 1.)
        alpha = xp.log(xp.sum(obs) / xp.sum(xp.exp(beta*i)))
        expected_counts = xp.exp(alpha + beta*i)

        # `table4` holds just the second and third columns from Table 4.
        table4 = xp.concat((obs[xp.newaxis, :],
                            expected_counts[xp.newaxis, :])).T

        table5 = xp.asarray([
            # lambda, statistic
            -10.0, 72.2e3,
            -5.0, 28.9e1,
            -3.0, 65.6,
            -2.0, 40.6,
            -1.5, 34.0,
            -1.0, 29.5,
            -0.5, 26.5,
            0.0, 24.6,
            0.5, 23.4,
            0.67, 23.1,
            1.0, 22.7,
            1.5, 22.6,
            2.0, 22.9,
            3.0, 24.8,
            5.0, 35.5,
            10.0, 21.4e1,
            ])
        table5 = xp.reshape(table5, (-1, 2))

        for i in range(table5.shape[0]):
            lambda_, expected_stat = table5[i, 0], table5[i, 1]
            stat, p = stats.power_divergence(table4[:,0], table4[:,1],
                                             lambda_=lambda_)
            xp_assert_close(stat, expected_stat, rtol=5e-3)


@make_xp_test_case(stats.chisquare)
class TestChisquare:
    def test_chisquare_12282a(self, xp):
        # Currently `chisquare` is implemented via power_divergence
        # in case that ever changes, perform a basic test like
        # test_power_divergence_gh_12282
        with assert_raises(ValueError, match='For each axis slice...'):
            f_obs = xp.asarray([10., 20.])
            f_exp = xp.asarray([30., 60.])
            stats.chisquare(f_obs, f_exp=f_exp)

    def test_chisquare_12282b(self, xp):
        # Check that users can now disable the sum check tested in
        # test_chisquare_12282a. Also, confirm that statistic and p-value
        # are as expected.
        rng = np.random.default_rng(3843874358728234)
        n = 10
        lam = rng.uniform(1000, 2000, size=n)
        x = rng.poisson(lam)
        lam = xp.asarray(lam)
        x = xp.asarray(x, dtype=lam.dtype)
        res = stats.chisquare(x, f_exp=lam, ddof=-1, sum_check=False)
        # Poisson is approximately normal with mean and variance lam
        z = (x - lam) / xp.sqrt(lam)
        statistic = xp.sum(z**2)
        xp_assert_close(res.statistic, statistic)
        # Sum of `n` squared standard normal variables follows chi2 with `n` DoF
        X2 = _SimpleChi2(xp.asarray(n, dtype=statistic.dtype))
        xp_assert_close(res.pvalue, X2.sf(statistic))

    @pytest.mark.parametrize("n, dtype", [(200, 'uint8'), (1000000, 'int32')])
    def test_chiquare_data_types_attributes(self, n, dtype, xp):
        # Regression test for gh-10159 and gh-18368
        dtype = getattr(xp, dtype)
        obs = xp.asarray([n, 0], dtype=dtype)
        exp = xp.asarray([n // 2, n // 2], dtype=dtype)
        res = stats.chisquare(obs, exp)
        stat, p = res
        xp_assert_close(stat, xp.asarray(n, dtype=xp.asarray(1.).dtype), rtol=1e-13)
        # check that attributes are identical to unpacked outputs - see gh-18368
        xp_assert_equal(res.statistic, stat)
        xp_assert_equal(res.pvalue, p)


@make_xp_test_case(stats.friedmanchisquare)
class TestFriedmanChiSquare:
    # verified with matlab and R
    # From Demsar "Statistical Comparisons of Classifiers over Multiple Data Sets"
    # 2006, Xf=9.28 (no tie handling, tie corrected Xf >=9.28)
    x1 = [[0.763, 0.599, 0.954, 0.628, 0.882, 0.936, 0.661,
           0.583, 0.775, 1.0, 0.94, 0.619, 0.972, 0.957],
          [0.768, 0.591, 0.971, 0.661, 0.888, 0.931, 0.668,
           0.583, 0.838, 1.0, 0.962, 0.666, 0.981, 0.978],
          [0.771, 0.590, 0.968, 0.654, 0.886, 0.916, 0.609,
           0.563, 0.866, 1.0, 0.965, 0.614, 0.9751, 0.946],
          [0.798, 0.569, 0.967, 0.657, 0.898, 0.931, 0.685,
           0.625, 0.875, 1.0, 0.962, 0.669, 0.975, 0.970]]
    ref1 = (10.2283464566929, 0.0167215803284414)

    # From "Bioestadistica para las ciencias de la salud" Xf=18.95 p<0.001:
    x2 = [[4, 3, 5, 3, 5, 3, 2, 5, 4, 4, 4, 3],
          [2, 2, 1, 2, 3, 1, 2, 3, 2, 1, 1, 3],
          [2, 4, 3, 3, 4, 3, 3, 4, 4, 1, 2, 1],
          [3, 5, 4, 3, 4, 4, 3, 3, 3, 4, 4, 4]]
    ref2 = (18.9428571428571, 0.000280938375189499)

    # From Jerrorl H. Zar, "Biostatistical Analysis"(example 12.6),
    # Xf=10.68, 0.005 < p < 0.01:
    # Probability from this example is inexact
    # using Chisquare approximation of Friedman Chisquare.
    x3 = [[7.0, 9.9, 8.5, 5.1, 10.3],
          [5.3, 5.7, 4.7, 3.5, 7.7],
          [4.9, 7.6, 5.5, 2.8, 8.4],
          [8.8, 8.9, 8.1, 3.3, 9.1]]
    ref3 = (10.68, 0.0135882729582176)

    @pytest.mark.parametrize("dtype", [None, "float32", "float64"])
    @pytest.mark.parametrize("data, ref", [(x1, ref1), (x2, ref2), (x3, ref3)])
    def test_against_references(self, dtype, data, ref, xp):
        if is_numpy(xp) and xp.__version__ < "2.0" and dtype=='float32':
            pytest.skip("NumPy doesn't preserve dtype pre-NEP 50.")
        dtype = dtype if dtype is None else getattr(xp, dtype)
        data = [xp.asarray(array, dtype=dtype) for array in data]
        res = stats.friedmanchisquare(*data)
        xp_assert_close(res.statistic, xp.asarray(ref[0], dtype=dtype))
        xp_assert_close(res.pvalue, xp.asarray(ref[1], dtype=dtype))

    def test_too_few_samples(self, xp):
        message = "At least 3 samples must be given"
        with pytest.raises(ValueError, match=message):
            stats.friedmanchisquare(xp.asarray(self.x3[0]), xp.asarray(self.x3[1]))


class TestKSTest:
    """Tests kstest and ks_1samp agree with K-S various sizes, alternatives, modes."""

    def _testOne(self, x, alternative, expected_statistic, expected_prob,
                 mode='auto', decimal=14):
        result = stats.kstest(x, 'norm', alternative=alternative, mode=mode)
        expected = np.array([expected_statistic, expected_prob])
        assert_array_almost_equal(np.array(result), expected, decimal=decimal)

    def _test_kstest_and_ks1samp(self, x, alternative, mode='auto', decimal=14):
        result = stats.kstest(x, 'norm', alternative=alternative, mode=mode)
        result_1samp = stats.ks_1samp(x, stats.norm.cdf,
                                      alternative=alternative, mode=mode)
        assert_array_almost_equal(np.array(result), result_1samp, decimal=decimal)

    def test_namedtuple_attributes(self):
        x = np.linspace(-1, 1, 9)
        # test for namedtuple attribute results
        attributes = ('statistic', 'pvalue')
        res = stats.kstest(x, 'norm')
        check_named_results(res, attributes)

    def test_agree_with_ks_1samp(self):
        x = np.linspace(-1, 1, 9)
        self._test_kstest_and_ks1samp(x, 'two-sided')

        x = np.linspace(-15, 15, 9)
        self._test_kstest_and_ks1samp(x, 'two-sided')

        x = [-1.23, 0.06, -0.60, 0.17, 0.66, -0.17, -0.08, 0.27, -0.98, -0.99]
        self._test_kstest_and_ks1samp(x, 'two-sided')
        self._test_kstest_and_ks1samp(x, 'greater', mode='exact')
        self._test_kstest_and_ks1samp(x, 'less', mode='exact')

    def test_pm_inf_gh20386(self):
        # Check that gh-20386 is resolved - `kstest` does not
        # return NaNs when both -inf and inf are in sample.
        vals = [-np.inf, 0, 1, np.inf]
        res = stats.kstest(vals, stats.cauchy.cdf)
        ref = stats.kstest(vals, stats.cauchy.cdf, _no_deco=True)
        assert np.all(np.isfinite(res))
        assert_equal(res, ref)
        assert not np.isnan(res.statistic)
        assert not np.isnan(res.pvalue)

    # missing: no test that uses *args


@make_xp_test_case(stats.ks_1samp)
class TestKSOneSample:
    """
    Tests kstest and ks_samp 1-samples with K-S various sizes, alternatives, modes.
    """

    def _testOne(self, x, alternative, expected_statistic, expected_prob, *,
                 mode='auto', dtype, xp):
        rtol = 5e-14 if dtype == xp.float64 else 1e-5
        res = stats.ks_1samp(x, special.ndtr, alternative=alternative, mode=mode)
        ref_statistic = xp.asarray(expected_statistic, dtype=dtype)
        ref_pvalue = xp.asarray(expected_prob, dtype=dtype)
        xp_assert_close(res.statistic, ref_statistic, rtol=rtol)
        xp_assert_close(res.pvalue, ref_pvalue, rtol=rtol)

    @pytest.mark.parametrize('dtype', [None, 'float32', 'float64'])
    def test_agree_with_r(self, dtype, xp):
        # comparing with some values from R
        if is_numpy(xp) and xp.__version__ < "2.0" and dtype == 'float32':
            pytest.skip("Pre-NEP 50 doesn't respect dtypes")
        dtype = xp_default_dtype(xp) if dtype is None else getattr(xp, dtype)
        x = xp.linspace(-1, 1, 9, dtype=dtype)
        self._testOne(x, 'two-sided', 0.15865525393145705, 0.95164069201518386,
                      dtype=dtype, xp=xp)

        x = xp.linspace(-15, 15, 9, dtype=dtype)
        self._testOne(x, 'two-sided', 0.44435602715924361, 0.038850140086788665,
                      dtype=dtype, xp=xp)

        x = [-1.23, 0.06, -0.60, 0.17, 0.66, -0.17, -0.08, 0.27, -0.98, -0.99]
        x = xp.asarray(x, dtype=dtype)
        self._testOne(x, 'two-sided', 0.293580126801961, 0.293408463684361,
                      dtype=dtype, xp=xp)
        self._testOne(x, 'greater', 0.293580126801961, 0.146988835042376, mode='exact',
                      dtype=dtype, xp=xp)
        self._testOne(x, 'less', 0.109348552425692, 0.732768892470675, mode='exact',
                      dtype=dtype, xp=xp)

    @pytest.mark.parametrize('dtype', [None, 'float32', 'float64'])
    def test_known_examples(self, xp, dtype):
        # the following tests rely on deterministically replicated rvs
        if is_numpy(xp) and xp.__version__ < "2.0" and dtype == 'float32':
            pytest.skip("Pre-NEP 50 doesn't respect dtypes")
        dtype = xp_default_dtype(xp) if dtype is None else getattr(xp, dtype)
        x = stats.norm.rvs(loc=0.2, size=100, random_state=987654321)
        x = xp.asarray(x, dtype=dtype)
        self._testOne(x, 'two-sided', 0.12464329735846891, 0.089444888711820769,
                      mode='asymp', xp=xp, dtype=dtype)
        self._testOne(x, 'less', 0.12464329735846891, 0.040989164077641749,
                      xp=xp, dtype=dtype)
        self._testOne(x, 'greater', 0.0072115233216310994, 0.98531158590396228,
                      xp=xp, dtype=dtype)

    # this is a test of the exact p-value calculation, available only with NumPy.
    def test_ks1samp_allpaths(self):
        # Check NaN input, output.
        assert_(np.isnan(kolmogn(np.nan, 1, True)))
        with assert_raises(ValueError, match='n is not integral: 1.5'):
            kolmogn(1.5, 1, True)
        assert_(np.isnan(kolmogn(-1, 1, True)))

        dataset = np.asarray([
            # Check x out of range
            (101, 1, True, 1.0),
            (101, 1.1, True, 1.0),
            (101, 0, True, 0.0),
            (101, -0.1, True, 0.0),

            (32, 1.0 / 64, True, 0.0),  # Ruben-Gambino
            (32, 1.0 / 64, False, 1.0),  # Ruben-Gambino

            # Miller
            (32, 0.5, True, 0.9999999363163307),
            # Miller 2 * special.smirnov(32, 0.5)
            (32, 0.5, False, 6.368366937916623e-08),

            # Check some other paths
            (32, 1.0 / 8, True, 0.34624229979775223),
            (32, 1.0 / 4, True, 0.9699508336558085),
            (1600, 0.49, False, 0.0),
            # 2 * special.smirnov(1600, 1/16.0)
            (1600, 1 / 16.0, False, 7.0837876229702195e-06),
            # _kolmogn_DMTW
            (1600, 14 / 1600, False, 0.99962357317602),
            # _kolmogn_PelzGood
            (1600, 1 / 32, False, 0.08603386296651416),
        ])
        FuncData(kolmogn, dataset, (0, 1, 2), 3).check(dtypes=[int, float, bool])

    @pytest.mark.parametrize("ksfunc", [stats.kstest, stats.ks_1samp])
    @pytest.mark.parametrize("alternative, x6val, ref_location, ref_sign",
                             [('greater', 6., 6., +1),
                              ('less', 7., 7., -1),
                              ('two-sided', 6., 6., +1),
                              ('two-sided', 7., 7., -1)])
    def test_location_sign(self, ksfunc, alternative,
                           x6val, ref_location, ref_sign, xp):
        # Test that location and sign corresponding with statistic are as
        # expected. (Test is designed to be easy to predict.)
        x = xp.arange(10.) + 0.5
        x = xpx.at(x)[6].set(x6val)
        # cdf = stats.uniform(scale=10).cdf
        def cdf(x): return x / 10.
        res = ksfunc(xp.asarray(x), cdf, alternative=alternative)
        rtol = 1e-15 if x.dtype == xp.float64 else 1e-6
        xp_assert_close(res.statistic, xp.asarray(0.1), rtol=rtol)
        xp_assert_equal(res.statistic_location, xp.asarray(ref_location))
        xp_assert_equal(res.statistic_sign, xp.asarray(ref_sign, dtype=xp.int8))

    # missing: no test that uses *args


class TestKSTwoSamples:
    """Tests 2-samples with K-S various sizes, alternatives, modes."""

    def _testOne(self, x1, x2, alternative, expected_statistic, expected_prob,
                 mode='auto'):
        result = stats.ks_2samp(x1, x2, alternative, mode=mode)
        expected = np.array([expected_statistic, expected_prob])
        assert_array_almost_equal(np.array(result), expected)

    def testSmall(self):
        self._testOne([0], [1], 'two-sided', 1.0/1, 1.0)
        self._testOne([0], [1], 'greater', 1.0/1, 0.5)
        self._testOne([0], [1], 'less', 0.0/1, 1.0)
        self._testOne([1], [0], 'two-sided', 1.0/1, 1.0)
        self._testOne([1], [0], 'greater', 0.0/1, 1.0)
        self._testOne([1], [0], 'less', 1.0/1, 0.5)

    def testTwoVsThree(self):
        data1 = np.array([1.0, 2.0])
        data1p = data1 + 0.01
        data1m = data1 - 0.01
        data2 = np.array([1.0, 2.0, 3.0])
        self._testOne(data1p, data2, 'two-sided', 1.0 / 3, 1.0)
        self._testOne(data1p, data2, 'greater', 1.0 / 3, 0.7)
        self._testOne(data1p, data2, 'less', 1.0 / 3, 0.7)
        self._testOne(data1m, data2, 'two-sided', 2.0 / 3, 0.6)
        self._testOne(data1m, data2, 'greater', 2.0 / 3, 0.3)
        self._testOne(data1m, data2, 'less', 0, 1.0)

    def testTwoVsFour(self):
        data1 = np.array([1.0, 2.0])
        data1p = data1 + 0.01
        data1m = data1 - 0.01
        data2 = np.array([1.0, 2.0, 3.0, 4.0])
        self._testOne(data1p, data2, 'two-sided', 2.0 / 4, 14.0/15)
        self._testOne(data1p, data2, 'greater', 2.0 / 4, 8.0/15)
        self._testOne(data1p, data2, 'less', 1.0 / 4, 12.0/15)

        self._testOne(data1m, data2, 'two-sided', 3.0 / 4, 6.0/15)
        self._testOne(data1m, data2, 'greater', 3.0 / 4, 3.0/15)
        self._testOne(data1m, data2, 'less', 0, 1.0)

    def test100_100(self):
        x100 = np.linspace(1, 100, 100)
        x100_2_p1 = x100 + 2 + 0.1
        x100_2_m1 = x100 + 2 - 0.1
        self._testOne(x100, x100_2_p1, 'two-sided', 3.0 / 100, 0.9999999999962055)
        self._testOne(x100, x100_2_p1, 'greater', 3.0 / 100, 0.9143290114276248)
        self._testOne(x100, x100_2_p1, 'less', 0, 1.0)
        self._testOne(x100, x100_2_m1, 'two-sided', 2.0 / 100, 1.0)
        self._testOne(x100, x100_2_m1, 'greater', 2.0 / 100, 0.960978450786184)
        self._testOne(x100, x100_2_m1, 'less', 0, 1.0)

    def test100_110(self):
        x100 = np.linspace(1, 100, 100)
        x110 = np.linspace(1, 100, 110)
        x110_20_p1 = x110 + 20 + 0.1
        x110_20_m1 = x110 + 20 - 0.1
        # 100, 110
        self._testOne(x100, x110_20_p1, 'two-sided', 232.0 / 1100, 0.015739183865607353)
        self._testOne(x100, x110_20_p1, 'greater', 232.0 / 1100, 0.007869594319053203)
        self._testOne(x100, x110_20_p1, 'less', 0, 1)
        self._testOne(x100, x110_20_m1, 'two-sided', 229.0 / 1100, 0.017803803861026313)
        self._testOne(x100, x110_20_m1, 'greater', 229.0 / 1100, 0.008901905958245056)
        self._testOne(x100, x110_20_m1, 'less', 0.0, 1.0)

    def testRepeatedValues(self):
        x2233 = np.array([2] * 3 + [3] * 4 + [5] * 5 + [6] * 4, dtype=int)
        x3344 = x2233 + 1
        x2356 = np.array([2] * 3 + [3] * 4 + [5] * 10 + [6] * 4, dtype=int)
        x3467 = np.array([3] * 10 + [4] * 2 + [6] * 10 + [7] * 4, dtype=int)
        self._testOne(x2233, x3344, 'two-sided', 5.0/16, 0.4262934613454952)
        self._testOne(x2233, x3344, 'greater', 5.0/16, 0.21465428276573786)
        self._testOne(x2233, x3344, 'less', 0.0/16, 1.0)
        self._testOne(x2356, x3467, 'two-sided', 190.0/21/26, 0.0919245790168125)
        self._testOne(x2356, x3467, 'greater', 190.0/21/26, 0.0459633806858544)
        self._testOne(x2356, x3467, 'less', 70.0/21/26, 0.6121593130022775)

    def testEqualSizes(self):
        data2 = np.array([1.0, 2.0, 3.0])
        self._testOne(data2, data2+1, 'two-sided', 1.0/3, 1.0)
        self._testOne(data2, data2+1, 'greater', 1.0/3, 0.75)
        self._testOne(data2, data2+1, 'less', 0.0/3, 1.)
        self._testOne(data2, data2+0.5, 'two-sided', 1.0/3, 1.0)
        self._testOne(data2, data2+0.5, 'greater', 1.0/3, 0.75)
        self._testOne(data2, data2+0.5, 'less', 0.0/3, 1.)
        self._testOne(data2, data2-0.5, 'two-sided', 1.0/3, 1.0)
        self._testOne(data2, data2-0.5, 'greater', 0.0/3, 1.0)
        self._testOne(data2, data2-0.5, 'less', 1.0/3, 0.75)

    @pytest.mark.slow
    def testMiddlingBoth(self):
        # 500, 600
        n1, n2 = 500, 600
        delta = 1.0/n1/n2/2/2
        x = np.linspace(1, 200, n1) - delta
        y = np.linspace(2, 200, n2)
        self._testOne(x, y, 'two-sided', 2000.0 / n1 / n2, 1.0,
                      mode='auto')
        self._testOne(x, y, 'two-sided', 2000.0 / n1 / n2, 1.0,
                      mode='asymp')
        self._testOne(x, y, 'greater', 2000.0 / n1 / n2, 0.9697596024683929,
                      mode='asymp')
        self._testOne(x, y, 'less', 500.0 / n1 / n2, 0.9968735843165021,
                      mode='asymp')
        with warnings.catch_warnings():
            message = "ks_2samp: Exact calculation unsuccessful."
            warnings.filterwarnings("ignore", message, RuntimeWarning)
            self._testOne(x, y, 'greater', 2000.0 / n1 / n2, 0.9697596024683929,
                          mode='exact')
            self._testOne(x, y, 'less', 500.0 / n1 / n2, 0.9968735843165021,
                          mode='exact')
        with warnings.catch_warnings(record=True) as w:
            warnings.simplefilter("always")
            self._testOne(x, y, 'less', 500.0 / n1 / n2, 0.9968735843165021,
                          mode='exact')
            _check_warnings(w, RuntimeWarning, 1)

    @pytest.mark.slow
    def testMediumBoth(self):
        # 1000, 1100
        n1, n2 = 1000, 1100
        delta = 1.0/n1/n2/2/2
        x = np.linspace(1, 200, n1) - delta
        y = np.linspace(2, 200, n2)
        self._testOne(x, y, 'two-sided', 6600.0 / n1 / n2, 1.0,
                      mode='asymp')
        self._testOne(x, y, 'two-sided', 6600.0 / n1 / n2, 1.0,
                      mode='auto')
        self._testOne(x, y, 'greater', 6600.0 / n1 / n2, 0.9573185808092622,
                      mode='asymp')
        self._testOne(x, y, 'less', 1000.0 / n1 / n2, 0.9982410869433984,
                      mode='asymp')

        with warnings.catch_warnings():
            message = "ks_2samp: Exact calculation unsuccessful."
            warnings.filterwarnings("ignore", message, RuntimeWarning)
            self._testOne(x, y, 'greater', 6600.0 / n1 / n2, 0.9573185808092622,
                          mode='exact')
            self._testOne(x, y, 'less', 1000.0 / n1 / n2, 0.9982410869433984,
                          mode='exact')
        with warnings.catch_warnings(record=True) as w:
            warnings.simplefilter("always")
            self._testOne(x, y, 'less', 1000.0 / n1 / n2, 0.9982410869433984,
                          mode='exact')
            _check_warnings(w, RuntimeWarning, 1)

    def testLarge(self):
        # 10000, 110
        n1, n2 = 10000, 110
        lcm = n1*11.0
        delta = 1.0/n1/n2/2/2
        x = np.linspace(1, 200, n1) - delta
        y = np.linspace(2, 100, n2)
        self._testOne(x, y, 'two-sided', 55275.0 / lcm, 4.2188474935755949e-15)
        self._testOne(x, y, 'greater', 561.0 / lcm, 0.99115454582047591)
        self._testOne(x, y, 'less', 55275.0 / lcm, 3.1317328311518713e-26)

    def test_gh11184(self):
        # 3000, 3001, exact two-sided
        rng = np.random.RandomState(123456)
        x = rng.normal(size=3000)
        y = rng.normal(size=3001) * 1.5
        self._testOne(x, y, 'two-sided', 0.11292880151060758, 2.7755575615628914e-15,
                      mode='asymp')
        self._testOne(x, y, 'two-sided', 0.11292880151060758, 2.7755575615628914e-15,
                      mode='exact')

    @pytest.mark.xslow
    def test_gh11184_bigger(self):
        # 10000, 10001, exact two-sided
        rng = np.random.RandomState(123456)
        x = rng.normal(size=10000)
        y = rng.normal(size=10001) * 1.5
        self._testOne(x, y, 'two-sided', 0.10597913208679133, 3.3149311398483503e-49,
                      mode='asymp')
        self._testOne(x, y, 'two-sided', 0.10597913208679133, 2.7755575615628914e-15,
                      mode='exact')
        self._testOne(x, y, 'greater', 0.10597913208679133, 2.7947433906389253e-41,
                      mode='asymp')
        self._testOne(x, y, 'less', 0.09658002199780022, 2.7947433906389253e-41,
                      mode='asymp')

    @pytest.mark.xslow
    def test_gh12999(self):
        rng = np.random.RandomState(123456)
        for x in range(1000, 12000, 1000):
            vals1 = rng.normal(size=(x))
            vals2 = rng.normal(size=(x + 10), loc=0.5)
            exact = stats.ks_2samp(vals1, vals2, mode='exact').pvalue
            asymp = stats.ks_2samp(vals1, vals2, mode='asymp').pvalue
            # these two p-values should be in line with each other
            assert_array_less(exact, 3 * asymp)
            assert_array_less(asymp, 3 * exact)

    @pytest.mark.slow
    def testLargeBoth(self):
        # 10000, 11000
        n1, n2 = 10000, 11000
        lcm = n1*11.0
        delta = 1.0/n1/n2/2/2
        x = np.linspace(1, 200, n1) - delta
        y = np.linspace(2, 200, n2)
        self._testOne(x, y, 'two-sided', 563.0 / lcm, 0.9990660108966576,
                      mode='asymp')
        self._testOne(x, y, 'two-sided', 563.0 / lcm, 0.9990456491488628,
                      mode='exact')
        self._testOne(x, y, 'two-sided', 563.0 / lcm, 0.9990660108966576,
                      mode='auto')
        self._testOne(x, y, 'greater', 563.0 / lcm, 0.7561851877420673)
        self._testOne(x, y, 'less', 10.0 / lcm, 0.9998239693191724)
        with warnings.catch_warnings():
            message = "ks_2samp: Exact calculation unsuccessful."
            warnings.filterwarnings("ignore", message, RuntimeWarning)
            self._testOne(x, y, 'greater', 563.0 / lcm, 0.7561851877420673,
                          mode='exact')
            self._testOne(x, y, 'less', 10.0 / lcm, 0.9998239693191724,
                          mode='exact')

    def testNamedAttributes(self):
        # test for namedtuple attribute results
        attributes = ('statistic', 'pvalue')
        res = stats.ks_2samp([1, 2], [3])
        check_named_results(res, attributes)

    @pytest.mark.slow
    def test_some_code_paths(self):
        # Check that some code paths are executed
        from scipy.stats._stats_py import (
            _count_paths_outside_method,
            _compute_outer_prob_inside_method
        )

        _compute_outer_prob_inside_method(1, 1, 1, 1)
        _count_paths_outside_method(1000, 1, 1, 1001)

        with np.errstate(invalid='raise'):
            assert_raises(FloatingPointError, _count_paths_outside_method,
                          1100, 1099, 1, 1)
            assert_raises(FloatingPointError, _count_paths_outside_method,
                          2000, 1000, 1, 1)

    @pytest.mark.parametrize('case', (([], [1]), ([1], []), ([], [])))
    def test_argument_checking(self, case):
        # Check that an empty array warns
        with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
            res = stats.ks_2samp(*case)
            assert_equal(res.statistic, np.nan)
            assert_equal(res.pvalue, np.nan)

    @pytest.mark.xslow
    def test_gh12218(self):
        """Ensure gh-12218 is fixed."""
        # gh-1228 triggered a TypeError calculating sqrt(n1*n2*(n1+n2)).
        # n1, n2 both large integers, the product exceeded 2^64
        rng = np.random.default_rng(8751495592)
        n1 = 2097152  # 2*^21
        rvs1 = stats.uniform.rvs(size=n1, loc=0., scale=1, random_state=rng)
        rvs2 = rvs1 + 1  # Exact value of rvs2 doesn't matter.
        stats.ks_2samp(rvs1, rvs2, alternative='greater', mode='asymp')
        stats.ks_2samp(rvs1, rvs2, alternative='less', mode='asymp')
        stats.ks_2samp(rvs1, rvs2, alternative='two-sided', mode='asymp')

    def test_warnings_gh_14019(self):
        # Check that RuntimeWarning is raised when method='auto' and exact
        # p-value calculation fails. See gh-14019.
        rng = np.random.RandomState(seed=23493549)
        # random samples of the same size as in the issue
        data1 = rng.random(size=881) + 0.5
        data2 = rng.random(size=369)
        message = "ks_2samp: Exact calculation unsuccessful"
        with pytest.warns(RuntimeWarning, match=message):
            res = stats.ks_2samp(data1, data2, alternative='less')
            assert_allclose(res.pvalue, 0, atol=1e-14)

    @pytest.mark.parametrize("ksfunc", [stats.kstest, stats.ks_2samp])
    @pytest.mark.parametrize("alternative, x6val, ref_location, ref_sign",
                             [('greater', 5.9, 5.9, +1),
                              ('less', 6.1, 6.0, -1),
                              ('two-sided', 5.9, 5.9, +1),
                              ('two-sided', 6.1, 6.0, -1)])
    def test_location_sign(self, ksfunc, alternative,
                           x6val, ref_location, ref_sign):
        # Test that location and sign corresponding with statistic are as
        # expected. (Test is designed to be easy to predict.)
        x = np.arange(10, dtype=np.float64)
        y = x.copy()
        x[6] = x6val
        res = stats.ks_2samp(x, y, alternative=alternative)
        assert res.statistic == 0.1
        assert res.statistic_location == ref_location
        assert res.statistic_sign == ref_sign


def test_ttest_rel():
    # regression test
    tr,pr = 0.81248591389165692, 0.41846234511362157
    tpr = ([tr,-tr],[pr,pr])

    rvs1 = np.linspace(1,100,100)
    rvs2 = np.linspace(1.01,99.989,100)
    rvs1_2D = np.array([np.linspace(1,100,100), np.linspace(1.01,99.989,100)])
    rvs2_2D = np.array([np.linspace(1.01,99.989,100), np.linspace(1,100,100)])

    t,p = stats.ttest_rel(rvs1, rvs2, axis=0)
    assert_array_almost_equal([t,p],(tr,pr))
    t,p = stats.ttest_rel(rvs1_2D.T, rvs2_2D.T, axis=0)
    assert_array_almost_equal([t,p],tpr)
    t,p = stats.ttest_rel(rvs1_2D, rvs2_2D, axis=1)
    assert_array_almost_equal([t,p],tpr)

    # test scalars
    with warnings.catch_warnings(), \
            np.errstate(invalid="ignore", divide="ignore"):
        warnings.filterwarnings(
            "ignore", "Degrees of freedom <= 0 for slice", RuntimeWarning)
        t, p = stats.ttest_rel(4., 3.)
    assert_(np.isnan(t))
    assert_(np.isnan(p))

    # test for namedtuple attribute results
    attributes = ('statistic', 'pvalue')
    res = stats.ttest_rel(rvs1, rvs2, axis=0)
    check_named_results(res, attributes)

    # test on 3 dimensions
    rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
    rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
    t,p = stats.ttest_rel(rvs1_3D, rvs2_3D, axis=1)
    assert_array_almost_equal(np.abs(t), tr)
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (2, 3))

    t, p = stats.ttest_rel(np.moveaxis(rvs1_3D, 2, 0),
                           np.moveaxis(rvs2_3D, 2, 0),
                           axis=2)
    assert_array_almost_equal(np.abs(t), tr)
    assert_array_almost_equal(np.abs(p), pr)
    assert_equal(t.shape, (3, 2))

    # test alternative parameter
    assert_raises(ValueError, stats.ttest_rel, rvs1, rvs2, alternative="error")

    t, p = stats.ttest_rel(rvs1, rvs2, axis=0, alternative="less")
    assert_allclose(p, 1 - pr/2)
    assert_allclose(t, tr)

    t, p = stats.ttest_rel(rvs1, rvs2, axis=0, alternative="greater")
    assert_allclose(p, pr/2)
    assert_allclose(t, tr)

    # check nan policy
    rng = np.random.RandomState(12345678)
    x = stats.norm.rvs(loc=5, scale=10, size=501, random_state=rng)
    x[500] = np.nan
    y = (stats.norm.rvs(loc=5, scale=10, size=501, random_state=rng) +
         stats.norm.rvs(scale=0.2, size=501, random_state=rng))
    y[500] = np.nan

    with np.errstate(invalid="ignore"):
        assert_array_equal(stats.ttest_rel(x, x), (np.nan, np.nan))

    assert_array_almost_equal(stats.ttest_rel(x, y, nan_policy='omit'),
                              (0.25299925303978066, 0.8003729814201519))
    assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='raise')
    assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='foobar')

    # test zero division problem
    with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
        t, p = stats.ttest_rel([0, 0, 0], [1, 1, 1])
    assert_equal((np.abs(t), p), (np.inf, 0))
    with np.errstate(invalid="ignore"):
        assert_equal(stats.ttest_rel([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))

        # check that nan in input array result in nan output
        anan = np.array([[1, np.nan], [-1, 1]])
        assert_equal(stats.ttest_rel(anan, np.zeros((2, 2))),
                     ([0, np.nan], [1, np.nan]))

    # test incorrect input shape raise an error
    x = np.arange(24)
    assert_raises(ValueError, stats.ttest_rel, x.reshape((8, 3)),
                  x.reshape((2, 3, 4)))

    # Convert from two-sided p-values to one sided using T result data.
    def convert(t, p, alt):
        if (t < 0 and alt == "less") or (t > 0 and alt == "greater"):
            return p / 2
        return 1 - (p / 2)
    converter = np.vectorize(convert)

    rvs1_2D[:, 20:30] = np.nan
    rvs2_2D[:, 15:25] = np.nan

    with pytest.warns(SmallSampleWarning, match=too_small_nd_omit):
        tr, pr = stats.ttest_rel(rvs1_2D, rvs2_2D, 0, nan_policy='omit')

    with pytest.warns(SmallSampleWarning, match=too_small_nd_omit):
        t, p = stats.ttest_rel(rvs1_2D, rvs2_2D, 0,
                               nan_policy='omit', alternative='less')
    assert_allclose(t, tr, rtol=1e-14)
    with np.errstate(invalid='ignore'):
        assert_allclose(p, converter(tr, pr, 'less'), rtol=1e-14)

    with pytest.warns(SmallSampleWarning, match=too_small_nd_omit):
        t, p = stats.ttest_rel(rvs1_2D, rvs2_2D, 0,
                               nan_policy='omit', alternative='greater')
    assert_allclose(t, tr, rtol=1e-14)
    with np.errstate(invalid='ignore'):
        assert_allclose(p, converter(tr, pr, 'greater'), rtol=1e-14)


def test_ttest_rel_nan_2nd_arg():
    # regression test for gh-6134: nans in the second arg were not handled
    x = [np.nan, 2.0, 3.0, 4.0]
    y = [1.0, 2.0, 1.0, 2.0]

    r1 = stats.ttest_rel(x, y, nan_policy='omit')
    r2 = stats.ttest_rel(y, x, nan_policy='omit')
    assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
    assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)

    # NB: arguments are paired when NaNs are dropped
    r3 = stats.ttest_rel(y[1:], x[1:])
    assert_allclose(r2, r3, atol=1e-15)

    # .. and this is consistent with R. R code:
    # x = c(NA, 2.0, 3.0, 4.0)
    # y = c(1.0, 2.0, 1.0, 2.0)
    # t.test(x, y, paired=TRUE)
    assert_allclose(r2, (-2, 0.1835), atol=1e-4)


def test_ttest_rel_empty_1d_returns_nan():
    # Two empty inputs should return a TtestResult containing nan
    # for both values.
    with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
        result = stats.ttest_rel([], [])
    assert isinstance(result, stats._stats_py.TtestResult)
    assert_equal(result, (np.nan, np.nan))


@pytest.mark.parametrize('b, expected_shape',
                         [(np.empty((1, 5, 0)), (3, 5)),
                          (np.empty((1, 0, 0)), (3, 0))])
def test_ttest_rel_axis_size_zero(b, expected_shape):
    # In this test, the length of the axis dimension is zero.
    # The results should be arrays containing nan with shape
    # given by the broadcast nonaxis dimensions.
    a = np.empty((3, 1, 0))
    with warnings.catch_warnings():
        # first case should warn, second shouldn't?
        warnings.filterwarnings("ignore", too_small_nd_not_omit, SmallSampleWarning)
        result = stats.ttest_rel(a, b, axis=-1)
    assert isinstance(result, stats._stats_py.TtestResult)
    expected_value = np.full(expected_shape, fill_value=np.nan)
    assert_equal(result.statistic, expected_value)
    assert_equal(result.pvalue, expected_value)


def test_ttest_rel_nonaxis_size_zero():
    # In this test, the length of the axis dimension is nonzero,
    # but one of the nonaxis dimensions has length 0.  Check that
    # we still get the correctly broadcast shape, which is (5, 0)
    # in this case.
    a = np.empty((1, 8, 0))
    b = np.empty((5, 8, 1))
    result = stats.ttest_rel(a, b, axis=1)
    assert isinstance(result, stats._stats_py.TtestResult)
    assert_equal(result.statistic.shape, (5, 0))
    assert_equal(result.pvalue.shape, (5, 0))


@pytest.mark.parametrize("alternative", ['two-sided', 'less', 'greater'])
def test_ttest_rel_ci_1d(alternative):
    # test confidence interval method against reference values
    rng = np.random.default_rng(3749065329432213059)
    n = 10
    x = rng.normal(size=n, loc=1.5, scale=2)
    y = rng.normal(size=n, loc=2, scale=2)
    # Reference values generated with R t.test:
    # options(digits=16)
    # x = c(1.22825792,  1.63950485,  4.39025641,  0.68609437,  2.03813481,
    #       -1.20040109,  1.81997937,  1.86854636,  2.94694282,  3.94291373)
    # y = c(3.49961496, 1.53192536, 5.53620083, 2.91687718, 0.04858043,
    #       3.78505943, 3.3077496 , 2.30468892, 3.42168074, 0.56797592)
    # t.test(x, y, paired=TRUE, conf.level=0.85, alternative='l')

    ref = {'two-sided': [-1.912194489914035, 0.400169725914035],
           'greater': [-1.563944820311475, np.inf],
           'less': [-np.inf, 0.05192005631147523]}
    res = stats.ttest_rel(x, y, alternative=alternative)
    ci = res.confidence_interval(confidence_level=0.85)
    assert_allclose(ci, ref[alternative])
    assert_equal(res.df, n-1)


@pytest.mark.parametrize("test_fun, args",
                         [(stats.ttest_1samp, (np.arange(10), 0)),
                          (stats.ttest_rel, (np.arange(10), np.arange(10)))])
def test_ttest_ci_iv(test_fun, args):
    # test `confidence_interval` method input validation
    res = test_fun(*args)
    message = '`confidence_level` must be a number between 0 and 1.'
    with pytest.raises(ValueError, match=message):
        res.confidence_interval(confidence_level=10)


def _desc_stats(x1, x2, axis=0, *, xp=None):
    xp = array_namespace(x1, x2) if xp is None else xp

    def _stats(x, axis=0):
        x = xp.asarray(x)
        mu = xp.mean(x, axis=axis)
        std = xp.std(x, axis=axis, correction=1)
        nobs = x.shape[axis]
        return mu, std, nobs

    return _stats(x1, axis) + _stats(x2, axis)


@make_xp_test_case(stats.ttest_ind, stats.ttest_ind_from_stats)
def test_ttest_ind(xp):
    # regression test
    tr = xp.asarray(1.0912746897927283)
    pr = xp.asarray(0.27647818616351882)
    tr_2D = xp.stack([tr, -tr])
    pr_2D = xp.stack([pr, pr])

    rvs1 = xp.linspace(5, 105, 100)
    rvs2 = xp.linspace(1, 100, 100)
    rvs1_2D = xp.stack([rvs1, rvs2])
    rvs2_2D = xp.stack([rvs2, rvs1])

    res = stats.ttest_ind(rvs1, rvs2, axis=0)
    t, p = res  # check that result object can be unpacked
    xp_assert_close(t, tr)
    xp_assert_close(p, pr)

    res = stats.ttest_ind_from_stats(*_desc_stats(rvs1, rvs2))
    t, p = res  # check that result object can be unpacked
    xp_assert_close(t, tr)
    xp_assert_close(p, pr)

    res = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0)
    xp_assert_close(res.statistic, tr_2D)
    xp_assert_close(res.pvalue, pr_2D)

    res = stats.ttest_ind_from_stats(*_desc_stats(rvs1_2D.T, rvs2_2D.T))
    xp_assert_close(res.statistic, tr_2D)
    xp_assert_close(res.pvalue, pr_2D)

    res = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1)
    xp_assert_close(res.statistic, tr_2D)
    xp_assert_close(res.pvalue, pr_2D)

    res = stats.ttest_ind_from_stats(*_desc_stats(rvs1_2D, rvs2_2D, axis=1))
    xp_assert_close(res.statistic, tr_2D)
    xp_assert_close(res.pvalue, pr_2D)

    # test on 3 dimensions removed because generic tests in
    # test_axis_nan_policy are much stronger

    # test alternative parameter
    message = "`alternative` must be 'less', 'greater', or 'two-sided'."
    with pytest.raises(ValueError, match=message):
        stats.ttest_ind(rvs1, rvs2, alternative = "error")

    args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
    with pytest.raises(ValueError, match=message):
        stats.ttest_ind_from_stats(*args, alternative = "error")

    t, p = stats.ttest_ind(rvs1, rvs2, alternative="less")
    xp_assert_close(p, 1 - (pr/2))
    xp_assert_close(t, tr)

    t, p = stats.ttest_ind(rvs1, rvs2, alternative="greater")
    xp_assert_close(p, pr/2)
    xp_assert_close(t, tr)

    # Check that ttest_ind_from_stats agrees with ttest_ind
    res1 = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, alternative="less")
    args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
    res2 = stats.ttest_ind_from_stats(*args, alternative="less")
    xp_assert_close(res1.statistic, res2.statistic)
    xp_assert_close(res1.pvalue, res2.pvalue)

    res1 = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, alternative="less")
    args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
    res2 = stats.ttest_ind_from_stats(*args, alternative="less")
    xp_assert_close(res1.statistic, res2.statistic)
    xp_assert_close(res1.pvalue, res2.pvalue)

    # test NaNs
    NaN = xp.asarray(xp.nan)
    rvs1 = xp.where(xp.arange(rvs1.shape[0]) == 0, NaN, rvs1)

    res = stats.ttest_ind(rvs1, rvs2, axis=0)
    xp_assert_equal(res.statistic, NaN)
    xp_assert_equal(res.pvalue, NaN)

    res = stats.ttest_ind_from_stats(*_desc_stats(rvs1, rvs2))
    xp_assert_equal(res.statistic, NaN)
    xp_assert_equal(res.pvalue, NaN)


def test_ttest_ind_nan_policy():
    rvs1 = np.linspace(5, 105, 100)
    rvs2 = np.linspace(1, 100, 100)
    rvs1_2D = np.array([rvs1, rvs2])
    rvs2_2D = np.array([rvs2, rvs1])
    rvs1_3D = np.dstack([rvs1_2D, rvs1_2D, rvs1_2D])
    rvs2_3D = np.dstack([rvs2_2D, rvs2_2D, rvs2_2D])

    # check nan policy
    rng = np.random.RandomState(12345678)
    x = stats.norm.rvs(loc=5, scale=10, size=501, random_state=rng)
    x[500] = np.nan
    y = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng)

    with np.errstate(invalid="ignore"):
        assert_array_equal(stats.ttest_ind(x, y), (np.nan, np.nan))

    assert_array_almost_equal(stats.ttest_ind(x, y, nan_policy='omit'),
                              (0.24779670949091914, 0.80434267337517906))
    assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='raise')
    assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='foobar')

    # test zero division problem
    with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
        t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1])
    assert_equal((np.abs(t), p), (np.inf, 0))

    with np.errstate(invalid="ignore"):
        assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))

        # check that nan in input array result in nan output
        anan = np.array([[1, np.nan], [-1, 1]])
        assert_equal(stats.ttest_ind(anan, np.zeros((2, 2))),
                     ([0, np.nan], [1, np.nan]))

    rvs1_3D[:, :, 10:15] = np.nan
    rvs2_3D[:, :, 6:12] = np.nan

    # Convert from two-sided p-values to one sided using T result data.
    def convert(t, p, alt):
        if (t < 0 and alt == "less") or (t > 0 and alt == "greater"):
            return p / 2
        return 1 - (p / 2)
    converter = np.vectorize(convert)

    tr, pr = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=0, nan_policy='omit')

    t, p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=0, nan_policy='omit',
                           alternative='less')
    assert_allclose(t, tr, rtol=1e-14)
    assert_allclose(p, converter(tr, pr, 'less'), rtol=1e-14)

    t, p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=0, nan_policy='omit',
                           alternative='greater')
    assert_allclose(t, tr, rtol=1e-14)
    assert_allclose(p, converter(tr, pr, 'greater'), rtol=1e-14)


def test_ttest_ind_scalar():
    # test scalars
    with warnings.catch_warnings(), np.errstate(invalid="ignore"):
        warnings.filterwarnings(
            "ignore", "Degrees of freedom <= 0 for slice", RuntimeWarning)
        t, p = stats.ttest_ind(4., 3.)
    assert np.isnan(t)
    assert np.isnan(p)


@pytest.mark.filterwarnings("ignore:Arguments...:DeprecationWarning")
class Test_ttest_ind_permutations:
    N = 20

    # data for most tests
    rng = np.random.default_rng(169708062)
    a = np.vstack((np.arange(3*N//4), rng.random(3*N//4)))
    b = np.vstack((np.arange(N//4) + 100, rng.random(N//4)))

    # data for equal variance tests
    a2 = np.arange(10)
    b2 = np.arange(10) + 100

    # data for exact test
    a3 = [1, 2]
    b3 = [3, 4]

    # data for bigger test
    rvs1 = stats.norm.rvs(loc=5, scale=10,  # type: ignore
                          size=500, random_state=rng).reshape(100, 5).T
    rvs2 = stats.norm.rvs(loc=8, scale=20, size=100, random_state=rng)  # type: ignore

    p_d = [1/1001, (676+1)/1001]  # desired pvalues
    p_d_gen = [1/1001, (672 + 1)/1001]  # desired pvalues for Generator seed
    p_d_big = [(993+1)/1001, (685+1)/1001, (840+1)/1001,
               (955+1)/1001, (255+1)/1001]

    params = [
        (a, b, {"axis": 1}, p_d),                     # basic test
        (a.T, b.T, {'axis': 0}, p_d),                 # along axis 0
        (a[0, :], b[0, :], {'axis': None}, p_d[0]),   # 1d data
        (a[0, :].tolist(), b[0, :].tolist(), {'axis': None}, p_d[0]),
        # different seeds
        (a, b, {'random_state': 0, "axis": 1}, p_d),
        (a, b, {'random_state': np.random.RandomState(0), "axis": 1}, p_d),
        (a2, b2, {'equal_var': True}, 1/1001),  # equal variances
        (rvs1, rvs2, {'axis': -1, 'random_state': 0}, p_d_big),  # bigger test
        (a3, b3, {}, 1/3),  # exact test
        (a, b, {'random_state': np.random.default_rng(0), "axis": 1}, p_d_gen),
        ]

    @pytest.mark.parametrize("alternative", ['less', 'greater', 'two-sided'])
    @pytest.mark.parametrize("shape", [(12,), (2, 12)])
    def test_permutation_method(self, alternative, shape):
        rng = np.random.default_rng(2348934579834565)
        x = rng.random(size=shape)
        y = rng.random(size=13)

        kwargs = dict(n_resamples=999)

        # Use ttest_ind with `method`
        rng = np.random.default_rng(348934579834565)
        method = stats.PermutationMethod(rng=rng, **kwargs)
        res = stats.ttest_ind(x, y, axis=-1, alternative=alternative, method=method)

        # Use `permutation_test` directly
        def statistic(x, y, axis): return stats.ttest_ind(x, y, axis=axis).statistic
        rng =  np.random.default_rng(348934579834565)
        ref = stats.permutation_test((x, y), statistic, axis=-1, rng=rng,
                                     alternative=alternative, **kwargs)

        assert_equal(res.statistic, ref.statistic)
        assert_equal(res.pvalue, ref.pvalue)

        # Sanity check against theoretical t-test
        ref = stats.ttest_ind(x, y, axis=-1, alternative=alternative)
        assert_equal(res.statistic, ref.statistic)
        assert_allclose(res.pvalue, ref.pvalue, rtol=3e-2)

    @pytest.mark.parametrize("alternative", ['less', 'greater', 'two-sided'])
    @pytest.mark.parametrize("shape", [(12,), (2, 12)])
    def test_monte_carlo_method(self, alternative, shape):
        rng = np.random.default_rng(2348934579834565)
        x = rng.random(size=shape)
        y = rng.random(size=13)

        kwargs = dict(n_resamples=999)

        # Use `monte_carlo` directly
        def statistic(x, y, axis): return stats.ttest_ind(x, y, axis=axis).statistic
        rng = np.random.default_rng(348934579834565)
        rvs = [rng.standard_normal, rng.standard_normal]
        ref = stats.monte_carlo_test((x, y), rvs=rvs, statistic=statistic, axis=-1,
                                     alternative=alternative, **kwargs)

        # Use ttest_ind with `method`
        rng = np.random.default_rng(348934579834565)
        rvs = [rng.standard_normal, rng.standard_normal]
        method = stats.MonteCarloMethod(rvs=rvs, **kwargs)
        res = stats.ttest_ind(x, y, axis=-1, alternative=alternative, method=method)
        assert_equal(res.statistic, ref.statistic)
        assert_equal(res.pvalue, ref.pvalue)

        # Passing `rng` instead of `rvs`
        method = stats.MonteCarloMethod(rng=348934579834565, **kwargs)
        res = stats.ttest_ind(x, y, axis=-1, alternative=alternative, method=method)
        assert_equal(res.statistic, ref.statistic)
        assert_equal(res.pvalue, ref.pvalue)

        # Sanity check against theoretical t-test
        ref = stats.ttest_ind(x, y, axis=-1, alternative=alternative)
        assert_equal(res.statistic, ref.statistic)
        assert_allclose(res.pvalue, ref.pvalue, rtol=6e-2)

    def test_resampling_input_validation(self):
        message = "`method` must be an instance of `PermutationMethod`, an instance..."
        with pytest.raises(ValueError, match=message):
            stats.ttest_ind([1, 2, 3], [4, 5, 6], method='migratory')


class Test_ttest_ind_common:
    # for tests that are performed on variations of the t-test (e.g. trimmed)
    @pytest.mark.xslow()
    @pytest.mark.parametrize("kwds", [{'trim': .2}, {}],
                             ids=["trim", "basic"])
    @pytest.mark.parametrize('equal_var', [True, False],
                             ids=['equal_var', 'unequal_var'])
    def test_ttest_many_dims(self, kwds, equal_var):
        # Test that test works on many-dimensional arrays
        rng = np.random.default_rng(3815288136)
        a = rng.random((5, 4, 4, 7, 1, 6))
        b = rng.random((4, 1, 8, 2, 6))
        res = stats.ttest_ind(a, b, axis=-3, **kwds)

        # compare fully-vectorized t-test against t-test on smaller slice
        i, j, k = 2, 3, 1
        a2 = a[i, :, j, :, 0, :]
        b2 = b[:, 0, :, k, :]
        res2 = stats.ttest_ind(a2, b2, axis=-2, **kwds)
        assert_equal(res.statistic[i, :, j, k, :],
                     res2.statistic)
        assert_equal(res.pvalue[i, :, j, k, :],
                     res2.pvalue)

        # compare against t-test on one axis-slice at a time

        # manually broadcast with tile; move axis to end to simplify
        x = np.moveaxis(np.tile(a, (1, 1, 1, 1, 2, 1)), -3, -1)
        y = np.moveaxis(np.tile(b, (5, 1, 4, 1, 1, 1)), -3, -1)
        shape = x.shape[:-1]
        statistics = np.zeros(shape)
        pvalues = np.zeros(shape)
        for indices in product(*(range(i) for i in shape)):
            xi = x[indices]  # use tuple to index single axis slice
            yi = y[indices]
            res3 = stats.ttest_ind(xi, yi, axis=-1, **kwds)
            statistics[indices] = res3.statistic
            pvalues[indices] = res3.pvalue

        assert_allclose(statistics, res.statistic)
        assert_allclose(pvalues, res.pvalue)

    @pytest.mark.parametrize("kwds", [{'trim': .2}, {}],
                             ids=["trim", "basic"])
    @pytest.mark.parametrize("axis", [-1, 0])
    def test_nans_on_axis(self, kwds, axis):
        # confirm that with `nan_policy='propagate'`, NaN results are returned
        # on the correct location
        rng = np.random.default_rng(363836384995579937222)
        a = rng.integers(10, size=(5, 3, 10)).astype('float')
        b = rng.integers(10, size=(5, 3, 10)).astype('float')
        # set some indices in `a` and `b` to be `np.nan`.
        a[0][2][3] = np.nan
        b[2][0][6] = np.nan

        # arbitrarily use `np.sum` as a baseline for which indices should be
        # NaNs
        expected = np.isnan(np.sum(a + b, axis=axis))
        # multidimensional inputs to `t.sf(np.abs(t), df)` with NaNs on some
        # indices throws an warning. See issue gh-13844
        with warnings.catch_warnings(), np.errstate(invalid="ignore"):
            warnings.filterwarnings(
                "ignore", "invalid value encountered in less_equal", RuntimeWarning)
            warnings.filterwarnings("ignore", "Precision loss occurred", RuntimeWarning)
            res = stats.ttest_ind(a, b, axis=axis, **kwds)
        p_nans = np.isnan(res.pvalue)
        assert_array_equal(p_nans, expected)
        statistic_nans = np.isnan(res.statistic)
        assert_array_equal(statistic_nans, expected)


class Test_ttest_trim:
    params = [
        [[1, 2, 3], [1.1, 2.9, 4.2], 0.53619490753126731, -0.6864951273557258,
         .2],
        [[56, 128.6, 12, 123.8, 64.34, 78, 763.3], [1.1, 2.9, 4.2],
         0.00998909252078421, 4.591598691181999, .2],
        [[56, 128.6, 12, 123.8, 64.34, 78, 763.3], [1.1, 2.9, 4.2],
         0.10512380092302633, 2.832256715395378, .32],
        [[2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9],
         [6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1],
         0.002878909511344, -4.2461168970325, .2],
        [[-0.84504783, 0.13366078, 3.53601757, -0.62908581, 0.54119466,
          -1.16511574, -0.08836614, 1.18495416, 2.48028757, -1.58925028,
          -1.6706357, 0.3090472, -2.12258305, 0.3697304, -1.0415207,
          -0.57783497, -0.90997008, 1.09850192, 0.41270579, -1.4927376],
         [1.2725522, 1.1657899, 2.7509041, 1.2389013, -0.9490494, -1.0752459,
          1.1038576, 2.9912821, 3.5349111, 0.4171922, 1.0168959, -0.7625041,
          -0.4300008, 3.0431921, 1.6035947, 0.5285634, -0.7649405, 1.5575896,
          1.3670797, 1.1726023], 0.005293305834235, -3.0983317739483, .2]]

    @pytest.mark.parametrize("a,b,pr,tr,trim", params)
    def test_ttest_compare_r(self, a, b, pr, tr, trim):
        '''
        Using PairedData's yuen.t.test method. Something to note is that there
        are at least 3 R packages that come with a trimmed t-test method, and
        comparisons were made between them. It was found that PairedData's
        method's results match this method, SAS, and one of the other R
        methods. A notable discrepancy was the DescTools implementation of the
        function, which only sometimes agreed with SAS, WRS2, PairedData and
        this implementation. For this reason, most comparisons in R are made
        against PairedData's method.

        Rather than providing the input and output for all evaluations, here is
        a representative example:
        > library(PairedData)
        > a <- c(1, 2, 3)
        > b <- c(1.1, 2.9, 4.2)
        > options(digits=16)
        > yuen.t.test(a, b, tr=.2)

            Two-sample Yuen test, trim=0.2

        data:  x and y
        t = -0.68649512735573, df = 3.4104431643464, p-value = 0.5361949075313
        alternative hypothesis: true difference in trimmed means is not equal
        to 0
        95 percent confidence interval:
         -3.912777195645217  2.446110528978550
        sample estimates:
        trimmed mean of x trimmed mean of y
        2.000000000000000 2.73333333333333
        '''
        statistic, pvalue = stats.ttest_ind(a, b, trim=trim, equal_var=False)
        assert_allclose(statistic, tr, atol=1e-15)
        assert_allclose(pvalue, pr, atol=1e-15)

    def test_compare_SAS(self):
        # Source of the data used in this test:
        # https://support.sas.com/resources/papers/proceedings14/1660-2014.pdf
        a = [12, 14, 18, 25, 32, 44, 12, 14, 18, 25, 32, 44]
        b = [17, 22, 14, 12, 30, 29, 19, 17, 22, 14, 12, 30, 29, 19]
        # In this paper, a trimming percentage of 5% is used. However,
        # in their implementation, the number of values trimmed is rounded to
        # the nearest whole number. However, consistent with
        # `scipy.stats.trimmed_mean`, this test truncates to the lower
        # whole number. In this example, the paper notes that 1 value is
        # trimmed off of each side. 9% replicates this amount of trimming.
        statistic, pvalue = stats.ttest_ind(a, b, trim=.09, equal_var=False)
        assert_allclose(pvalue, 0.514522, atol=1e-6)
        assert_allclose(statistic, 0.669169, atol=1e-6)

    def test_equal_var(self):
        '''
        The PairedData library only supports unequal variances. To compare
        samples with equal variances, the multicon library is used.
        > library(multicon)
        > a <- c(2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9)
        > b <- c(6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1)
        > dv = c(a,b)
        > iv = c(rep('a', length(a)), rep('b', length(b)))
        > yuenContrast(dv~ iv, EQVAR = TRUE)
        $Ms
           N                 M wgt
        a 11 2.442857142857143   1
        b 11 5.385714285714286  -1

        $test
                              stat df              crit                   p
        results -4.246116897032513 12 2.178812829667228 0.00113508833897713
        '''
        a = [2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9]
        b = [6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1]
        # `equal_var=True` is default
        statistic, pvalue = stats.ttest_ind(a, b, trim=.2)
        assert_allclose(pvalue, 0.00113508833897713, atol=1e-10)
        assert_allclose(statistic, -4.246116897032513, atol=1e-10)

    @pytest.mark.parametrize('alt,pr,tr',
                             (('greater', 0.9985605452443, -4.2461168970325),
                              ('less', 0.001439454755672, -4.2461168970325),),
                             )
    def test_alternatives(self, alt, pr, tr):
        '''
        > library(PairedData)
        > a <- c(2.7,2.7,1.1,3.0,1.9,3.0,3.8,3.8,0.3,1.9,1.9)
        > b <- c(6.5,5.4,8.1,3.5,0.5,3.8,6.8,4.9,9.5,6.2,4.1)
        > options(digits=16)
        > yuen.t.test(a, b, alternative = 'greater')
        '''
        a = [2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9]
        b = [6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1]

        statistic, pvalue = stats.ttest_ind(a, b, trim=.2, equal_var=False,
                                            alternative=alt)
        assert_allclose(pvalue, pr, atol=1e-10)
        assert_allclose(statistic, tr, atol=1e-10)

    @skip_xp_backends(cpu_only=True, reason='Uses NumPy for pvalue, CI')
    def test_permutation_not_implement_for_xp(self, xp):
        message = "Use of `trim` is compatible only with NumPy arrays."
        a, b = xp.arange(10), xp.arange(10)+1
        if is_numpy(xp):  # no error
            stats.ttest_ind(a, b, trim=0.1)
        else:  # NotImplementedError
            with pytest.raises(NotImplementedError, match=message):
                stats.ttest_ind(a, b, trim=0.1)

    @pytest.mark.parametrize("trim", [-.2, .5, 1])
    def test_trim_bounds_error(self, trim):
        match = "Trimming percentage should be 0 <= `trim` < .5."
        with assert_raises(ValueError, match=match):
            stats.ttest_ind([1, 2], [2, 1], trim=trim)


@make_xp_test_case(stats.ttest_ind)
class Test_ttest_CI:
    # indices in order [alternative={two-sided, less, greater},
    #                   equal_var={False, True}, trim={0, 0.2}]
    # reference values in order `statistic, df, pvalue, low, high`
    # equal_var=False reference values computed with R PairedData yuen.t.test:
    #
    # library(PairedData)
    # options(digits=16)
    # a < - c(0.88236329, 0.97318744, 0.4549262, 0.97893335, 0.0606677,
    #         0.44013366, 0.55806018, 0.40151434, 0.14453315, 0.25860601,
    #         0.20202162)
    # b < - c(0.93455277, 0.42680603, 0.49751939, 0.14152846, 0.711435,
    #         0.77669667, 0.20507578, 0.78702772, 0.94691855, 0.32464958,
    #         0.3873582, 0.35187468, 0.21731811)
    # yuen.t.test(a, b, tr=0, conf.level = 0.9, alternative = 'l')
    #
    # equal_var=True reference values computed with R multicon yuenContrast:
    #
    # library(multicon)
    # options(digits=16)
    # a < - c(0.88236329, 0.97318744, 0.4549262, 0.97893335, 0.0606677,
    #         0.44013366, 0.55806018, 0.40151434, 0.14453315, 0.25860601,
    #         0.20202162)
    # b < - c(0.93455277, 0.42680603, 0.49751939, 0.14152846, 0.711435,
    #         0.77669667, 0.20507578, 0.78702772, 0.94691855, 0.32464958,
    #         0.3873582, 0.35187468, 0.21731811)
    # dv = c(a, b)
    # iv = c(rep('a', length(a)), rep('b', length(b)))
    # yuenContrast(dv~iv, EQVAR = FALSE, alternative = 'unequal', tr = 0.2)
    r = np.empty(shape=(3, 2, 2, 5))
    r[0, 0, 0] = [-0.2314607, 19.894435, 0.8193209, -0.247220294, 0.188729943]
    r[1, 0, 0] = [-0.2314607, 19.894435, 0.40966045, -np.inf, 0.1382426469]
    r[2, 0, 0] = [-0.2314607, 19.894435, 0.5903395, -0.1967329982, np.inf]
    r[0, 0, 1] = [-0.2452886, 11.427896, 0.8105823, -0.34057446, 0.25847383]
    r[1, 0, 1] = [-0.2452886, 11.427896, 0.40529115, -np.inf, 0.1865829074]
    r[2, 0, 1] = [-0.2452886, 11.427896, 0.5947089, -0.268683541, np.inf]
    # confidence interval not available for equal_var=True
    r[0, 1, 0] = [-0.2345625322555006, 22, 0.8167175905643815, np.nan, np.nan]
    r[1, 1, 0] = [-0.2345625322555006, 22, 0.4083587952821908, np.nan, np.nan]
    r[2, 1, 0] = [-0.2345625322555006, 22, 0.5916412047178092, np.nan, np.nan]
    r[0, 1, 1] = [-0.2505369406507428, 14, 0.8058115135702835, np.nan, np.nan]
    r[1, 1, 1] = [-0.2505369406507428, 14, 0.4029057567851417, np.nan, np.nan]
    r[2, 1, 1] = [-0.2505369406507428, 14, 0.5970942432148583, np.nan, np.nan]
    @pytest.mark.parametrize('alternative', ['two-sided', 'less', 'greater'])
    @pytest.mark.parametrize('equal_var', [False, True])
    @pytest.mark.parametrize('trim', [0, 0.2])
    @skip_xp_backends('jax.numpy', reason='Generic stdtrit mutates array.')
    def test_confidence_interval(self, alternative, equal_var, trim, xp):
        if equal_var and trim:
            pytest.xfail('Discrepancy in `main`; needs further investigation.')

        if trim and not is_numpy(xp):
            pytest.skip('`trim` is only compatible with NumPy input')

        rng = np.random.default_rng(3810954496107292580)
        x = xp.asarray(rng.random(11))
        y = xp.asarray(rng.random(13))

        res = stats.ttest_ind(x, y, alternative=alternative,
                              equal_var=equal_var, trim=trim)

        alternatives = {'two-sided': 0, 'less': 1, 'greater': 2}
        ref = self.r[alternatives[alternative], int(equal_var), int(np.ceil(trim))]
        statistic, df, pvalue, low, high = ref

        rtol = 1e-7  # only 7 digits in reference
        xp_assert_close(res.statistic, xp.asarray(statistic), rtol=rtol)
        xp_assert_close(res.df, xp.asarray(df), rtol=rtol)
        xp_assert_close(res.pvalue, xp.asarray(pvalue), rtol=rtol)

        if not equal_var:  # CI not available when `equal_var is True`
            ci = res.confidence_interval(0.9)
            xp_assert_close(ci.low, xp.asarray(low), rtol=rtol)
            xp_assert_close(ci.high, xp.asarray(high), rtol=rtol)


def test__broadcast_concatenate():
    # test that _broadcast_concatenate properly broadcasts arrays along all
    # axes except `axis`, then concatenates along axis
    rng = np.random.default_rng(7544340069)
    a = rng.random((5, 4, 4, 3, 1, 6))
    b = rng.random((4, 1, 8, 2, 6))
    c = _broadcast_concatenate((a, b), axis=-3)
    # broadcast manually as an independent check
    a = np.tile(a, (1, 1, 1, 1, 2, 1))
    b = np.tile(b[None, ...], (5, 1, 4, 1, 1, 1))
    for index in product(*(range(i) for i in c.shape)):
        i, j, k, l, m, n = index
        if l < a.shape[-3]:
            assert a[i, j, k, l, m, n] == c[i, j, k, l, m, n]
        else:
            assert b[i, j, k, l - a.shape[-3], m, n] == c[i, j, k, l, m, n]


@make_xp_test_case(stats.ttest_ind)
class TestTTestInd:
    @make_xp_test_case(stats.ttest_ind_from_stats)
    def test_ttest_ind_with_uneq_var(self, xp):
        # check vs. R `t.test`, e.g.
        # options(digits=20)
        # a = c(1., 2., 3.)
        # b = c(1.1, 2.9, 4.2)
        # t.test(a, b, equal.var=FALSE)

        a = xp.asarray([1., 2., 3.])
        b = xp.asarray([1.1, 2.9, 4.2])
        pr = xp.asarray(0.53619490753126686)
        tr = xp.asarray(-0.686495127355726265)

        t, p = stats.ttest_ind(a, b, equal_var=False)
        xp_assert_close(t, tr)
        xp_assert_close(p, pr)

        t, p = stats.ttest_ind_from_stats(*_desc_stats(a, b), equal_var=False)
        xp_assert_close(t, tr)
        xp_assert_close(p, pr)

        a = xp.asarray([1., 2., 3., 4.])
        pr = xp.asarray(0.84354139131608252)
        tr = xp.asarray(-0.210866331595072315)

        t, p = stats.ttest_ind(a, b, equal_var=False)
        xp_assert_close(t, tr)
        xp_assert_close(p, pr)

        t, p = stats.ttest_ind_from_stats(*_desc_stats(a, b), equal_var=False)
        xp_assert_close(t, tr)
        xp_assert_close(p, pr)

        # regression test
        tr = xp.asarray(1.0912746897927283)
        tr_uneq_n = xp.asarray(0.66745638708050492)
        pr = xp.asarray(0.27647831993021388)
        pr_uneq_n = xp.asarray(0.50873585065616544)
        tr_2D = xp.stack([tr, -tr])
        pr_2D = xp.stack([pr, pr])

        rvs3 = xp.linspace(1, 100, 25)
        rvs2 = xp.linspace(1, 100, 100)
        rvs1 = xp.linspace(5, 105, 100)
        rvs1_2D = xp.stack([rvs1, rvs2])
        rvs2_2D = xp.stack([rvs2, rvs1])

        t, p = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
        xp_assert_close(t, tr)
        xp_assert_close(p, pr)

        t, p = stats.ttest_ind_from_stats(*_desc_stats(rvs1, rvs2), equal_var=False)
        xp_assert_close(t, tr)
        xp_assert_close(p, pr)

        t, p = stats.ttest_ind(rvs1, rvs3, axis=0, equal_var=False)
        xp_assert_close(t, tr_uneq_n)
        xp_assert_close(p, pr_uneq_n)

        t, p = stats.ttest_ind_from_stats(*_desc_stats(rvs1, rvs3), equal_var=False)
        xp_assert_close(t, tr_uneq_n)
        xp_assert_close(p, pr_uneq_n)

        res = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, equal_var=False)
        xp_assert_close(res.statistic, tr_2D)
        xp_assert_close(res.pvalue, pr_2D)

        args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
        res = stats.ttest_ind_from_stats(*args, equal_var=False)
        xp_assert_close(res.statistic, tr_2D)
        xp_assert_close(res.pvalue, pr_2D)

        res = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1, equal_var=False)
        xp_assert_close(res.statistic, tr_2D)
        xp_assert_close(res.pvalue, pr_2D)

        args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
        res = stats.ttest_ind_from_stats(*args, equal_var=False)
        xp_assert_close(res.statistic, tr_2D)
        xp_assert_close(res.pvalue, pr_2D)

    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered:RuntimeWarning"
    ) # for dask
    @pytest.mark.filterwarnings(
        "ignore:invalid value encountered:RuntimeWarning"
    ) # for dask
    def test_ttest_ind_zero_division(self, xp):
        # test zero division problem
        x = xp.zeros(3)
        y = xp.ones(3)

        with eager_warns(RuntimeWarning, match="Precision loss occurred", xp=xp):
            t, p = stats.ttest_ind(x, y, equal_var=False)

        xp_assert_equal(t, xp.asarray(-xp.inf))
        xp_assert_equal(p, xp.asarray(0.))

        with np.errstate(all='ignore'):
            t, p = stats.ttest_ind(x, x, equal_var=False)
            xp_assert_equal(t, xp.asarray(xp.nan))
            xp_assert_equal(p, xp.asarray(xp.nan))

            # check that nan in input array result in nan output
            anan = xp.asarray([[1, xp.nan], [-1, 1]])
            t, p = stats.ttest_ind(anan, xp.zeros((2, 2)), equal_var=False)
            xp_assert_equal(t, xp.asarray([0., np.nan]))
            xp_assert_equal(p, xp.asarray([1., np.nan]))

    def test_ttest_ind_nan_2nd_arg(self):
        # regression test for gh-6134: nans in the second arg were not handled
        x = [np.nan, 2.0, 3.0, 4.0]
        y = [1.0, 2.0, 1.0, 2.0]

        r1 = stats.ttest_ind(x, y, nan_policy='omit')
        r2 = stats.ttest_ind(y, x, nan_policy='omit')
        assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
        assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)

        # NB: arguments are not paired when NaNs are dropped
        r3 = stats.ttest_ind(y, x[1:])
        assert_allclose(r2, r3, atol=1e-15)

        # .. and this is consistent with R. R code:
        # x = c(NA, 2.0, 3.0, 4.0)
        # y = c(1.0, 2.0, 1.0, 2.0)
        # t.test(x, y, var.equal=TRUE)
        assert_allclose(r2, (-2.5354627641855498, 0.052181400457057901),
                        atol=1e-15)

    def test_ttest_ind_empty_1d_returns_nan(self, xp):
        # Two empty inputs should return a TtestResult containing nan
        # for both values.
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            res = stats.ttest_ind(xp.asarray([]), xp.asarray([]))
        assert isinstance(res, stats._stats_py.TtestResult)
        NaN = xp.asarray(xp.nan)[()]
        xp_assert_equal(res.statistic, NaN)
        xp_assert_equal(res.pvalue, NaN)

    @pytest.mark.parametrize('b, expected_shape',
                            [(np.empty((1, 5, 0)), (3, 5)),
                            (np.empty((1, 0, 0)), (3, 0))])
    def test_ttest_ind_axis_size_zero(self, b, expected_shape, xp):
        # In this test, the length of the axis dimension is zero.
        # The results should be arrays containing nan with shape
        # given by the broadcast nonaxis dimensions.
        a = xp.empty((3, 1, 0))
        b = xp.asarray(b, dtype=a.dtype)
        with warnings.catch_warnings():
            # first case should warn, second shouldn't?
            warnings.filterwarnings("ignore", too_small_nd_not_omit, SmallSampleWarning)
            res = stats.ttest_ind(a, b, axis=-1)
        assert isinstance(res, stats._stats_py.TtestResult)
        expected_value = xp.full(expected_shape, fill_value=xp.nan)
        xp_assert_equal(res.statistic, expected_value)
        xp_assert_equal(res.pvalue, expected_value)

    def test_ttest_ind_nonaxis_size_zero(self, xp):
        # In this test, the length of the axis dimension is nonzero,
        # but one of the nonaxis dimensions has length 0.  Check that
        # we still get the correctly broadcast shape, which is (5, 0)
        # in this case.
        a = xp.empty((1, 8, 0))
        b = xp.empty((5, 8, 1))
        res = stats.ttest_ind(a, b, axis=1)
        assert isinstance(res, stats._stats_py.TtestResult)
        assert res.statistic.shape ==(5, 0)
        assert res.pvalue.shape == (5, 0)

    def test_ttest_ind_nonaxis_size_zero_different_lengths(self, xp):
        # In this test, the length of the axis dimension is nonzero,
        # and that size is different in the two inputs,
        # and one of the nonaxis dimensions has length 0.  Check that
        # we still get the correctly broadcast shape, which is (5, 0)
        # in this case.
        a = xp.empty((1, 7, 0))
        b = xp.empty((5, 8, 1))
        res = stats.ttest_ind(a, b, axis=1)
        assert isinstance(res, stats._stats_py.TtestResult)
        assert res.statistic.shape ==(5, 0)
        assert res.pvalue.shape == (5, 0)


@make_xp_test_case(stats.ttest_ind_from_stats)
class TestTTestIndFromStats:
    @pytest.mark.skip_xp_backends(np_only=True,
                                reason="Other backends don't like integers")
    def test_gh5686(self, xp):
        mean1, mean2 = xp.asarray([1, 2]), xp.asarray([3, 4])
        std1, std2 = xp.asarray([5, 3]), xp.asarray([4, 5])
        nobs1, nobs2 = xp.asarray([130, 140]), xp.asarray([100, 150])
        # This will raise a TypeError unless gh-5686 is fixed.
        stats.ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2)

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning")
    def test_ttest_ind_from_stats_inputs_zero(self, xp):
        # Regression test for gh-6409.
        zero = xp.asarray(0.)
        six = xp.asarray(6.)
        NaN = xp.asarray(xp.nan)
        res = stats.ttest_ind_from_stats(zero, zero, six, zero, zero, six,
                                         equal_var=False)
        xp_assert_equal(res.statistic, NaN)
        xp_assert_equal(res.pvalue, NaN)


@pytest.mark.skip_xp_backends(cpu_only=True, reason='Test uses ks_1samp')
@pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
@pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
def test_ttest_uniform_pvalues(xp):
    # test that p-values are uniformly distributed under the null hypothesis
    rng = np.random.default_rng(246834602926842)
    x = xp.asarray(rng.normal(size=(10000, 2)))
    y = xp.asarray(rng.normal(size=(10000, 1)))
    q = rng.uniform(size=100)

    res = stats.ttest_ind(x, y, equal_var=True, axis=-1)
    pvalue = np.asarray(res.pvalue)
    assert stats.ks_1samp(pvalue, stats.uniform().cdf).pvalue > 0.1
    assert_allclose(np.quantile(pvalue, q), q, atol=1e-2)

    res = stats.ttest_ind(y, x, equal_var=True, axis=-1)
    pvalue = np.asarray(res.pvalue)
    assert stats.ks_1samp(pvalue, stats.uniform().cdf).pvalue > 0.1
    assert_allclose(np.quantile(pvalue, q), q, atol=1e-2)

    # reference values from R:
    # options(digits=16)
    # t.test(c(2, 3, 5), c(1.5), var.equal=TRUE)
    x, y = xp.asarray([2., 3., 5.]), xp.asarray([1.5])

    res = stats.ttest_ind(x, y, equal_var=True)
    xp_assert_close(res.statistic, xp.asarray(1.0394023007754))
    xp_assert_close(res.pvalue, xp.asarray(0.407779907736))


def _convert_pvalue_alternative(t, p, alt, xp):
    # test alternative parameter
    # Convert from two-sided p-values to one sided using T result data.
    less = xp.asarray(alt == "less")
    greater = xp.asarray(alt == "greater")
    i = ((t < 0) & less) | ((t > 0) & greater)
    return xp.where(i, p/2, 1 - p/2)


@pytest.mark.slow
@pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
@pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
@make_xp_test_case(stats.ttest_1samp)
def test_ttest_1samp_new(xp):
    rng = np.random.default_rng(88123290)
    n1, n2, n3 = (10, 15, 20)
    rvn1 = stats.norm.rvs(loc=5, scale=10, size=(n1, n2, n3), random_state=rng)
    rvn1 = xp.asarray(rvn1)

    # check multidimensional array and correct axis handling
    # deterministic rvn1 and rvn2 would be better as in test_ttest_rel
    popmean = xp.ones((1, n2, n3))
    t1, p1 = stats.ttest_1samp(rvn1, popmean, axis=0)
    t2, p2 = stats.ttest_1samp(rvn1, 1., axis=0)
    t3, p3 = stats.ttest_1samp(rvn1[:, 0, 0], 1.)
    xp_assert_close(t1, t2, rtol=1e-14)
    xp_assert_close(t1[0, 0], t3, rtol=1e-14)
    assert_equal(t1.shape, (n2, n3))

    popmean = xp.ones((n1, 1, n3))
    t1, p1 = stats.ttest_1samp(rvn1, popmean, axis=1)
    t2, p2 = stats.ttest_1samp(rvn1, 1., axis=1)
    t3, p3 = stats.ttest_1samp(rvn1[0, :, 0], 1.)
    xp_assert_close(t1, t2, rtol=1e-14)
    xp_assert_close(t1[0, 0], t3, rtol=1e-14)
    assert_equal(t1.shape, (n1, n3))

    popmean = xp.ones((n1, n2, 1))
    t1, p1 = stats.ttest_1samp(rvn1, popmean, axis=2)
    t2, p2 = stats.ttest_1samp(rvn1, 1., axis=2)
    t3, p3 = stats.ttest_1samp(rvn1[0, 0, :], 1.)
    xp_assert_close(t1, t2, rtol=1e-14)
    xp_assert_close(t1[0, 0], t3, rtol=1e-14)
    assert_equal(t1.shape, (n1, n2))

    # test zero division problem
    t, p = stats.ttest_1samp(xp.asarray([0., 0., 0.]), 1.)
    xp_assert_equal(xp.abs(t), xp.asarray(xp.inf))
    xp_assert_equal(p, xp.asarray(0.))

    tr, pr = stats.ttest_1samp(rvn1[:, :, :], 1.)

    t, p = stats.ttest_1samp(rvn1[:, :, :], 1., alternative="greater")
    pc = _convert_pvalue_alternative(tr, pr, "greater", xp)
    xp_assert_close(p, pc)
    xp_assert_close(t, tr)

    t, p = stats.ttest_1samp(rvn1[:, :, :], 1., alternative="less")
    pc = _convert_pvalue_alternative(tr, pr, "less", xp)
    xp_assert_close(p, pc)
    xp_assert_close(t, tr)

    with np.errstate(all='ignore'):
        res = stats.ttest_1samp(xp.asarray([0., 0., 0.]), 0.)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

        # check that nan in input array result in nan output
        anan = xp.asarray([[1., np.nan], [-1., 1.]])
        res = stats.ttest_1samp(anan, 0.)
        xp_assert_equal(res.statistic, xp.asarray([0., xp.nan]))
        xp_assert_equal(res.pvalue, xp.asarray([1., xp.nan]))


@skip_xp_backends(np_only=True, reason="Only NumPy has nan_policy='omit' for now")
def test_ttest_1samp_new_omit(xp):
    rng = np.random.default_rng(4008400329)
    n1, n2, n3 = (5, 10, 15)
    rvn1 = stats.norm.rvs(loc=5, scale=10, size=(n1, n2, n3), random_state=rng)
    rvn1 = xp.asarray(rvn1)

    rvn1[0:2, 1:3, 4:8] = xp.nan

    tr, pr = stats.ttest_1samp(rvn1[:, :, :], 1., nan_policy='omit')

    t, p = stats.ttest_1samp(rvn1[:, :, :], 1., nan_policy='omit',
                             alternative="greater")
    pc = _convert_pvalue_alternative(tr, pr, "greater", xp)
    xp_assert_close(p, pc)
    xp_assert_close(t, tr)

    t, p = stats.ttest_1samp(rvn1[:, :, :], 1., nan_policy='omit',
                             alternative="less")
    pc = _convert_pvalue_alternative(tr, pr, "less", xp)
    xp_assert_close(p, pc)
    xp_assert_close(t, tr)


@make_xp_test_case(stats.ttest_1samp)
@pytest.mark.skip_xp_backends('jax.numpy', reason='Generic stdtrit mutates array.')
def test_ttest_1samp_popmean_array(xp):
    # when popmean.shape[axis] != 1, raise an error
    # if the user wants to test multiple null hypotheses simultaneously,
    # use standard broadcasting rules
    rng = np.random.default_rng(2913300596553337193)
    x = rng.random(size=(1, 15, 20))
    x = xp.asarray(x)

    message = r"`popmean.shape\[axis\]` must equal 1."
    popmean = xp.asarray(rng.random(size=(5, 2, 20)))
    with pytest.raises(ValueError, match=message):
        stats.ttest_1samp(x, popmean=popmean, axis=-2)

    popmean = xp.asarray(rng.random(size=(5, 1, 20)))
    res = stats.ttest_1samp(x, popmean=popmean, axis=-2)
    assert res.statistic.shape == (5, 20)

    l, u = res.confidence_interval()
    l = xp.expand_dims(l, axis=-2)
    u = xp.expand_dims(u, axis=-2)

    res = stats.ttest_1samp(x, popmean=l, axis=-2)
    ref = xp.broadcast_to(xp.asarray(0.05, dtype=xp.float64), res.pvalue.shape)
    xp_assert_close(res.pvalue, ref)

    res = stats.ttest_1samp(x, popmean=u, axis=-2)
    xp_assert_close(res.pvalue, ref)


@make_xp_test_case(stats.describe)
class TestDescribe:
    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_describe_scalar(self, xp):
        with warnings.catch_warnings(), \
              np.errstate(invalid="ignore", divide="ignore"):
            warnings.filterwarnings(
                "ignore", "Degrees of freedom <= 0 for slice", RuntimeWarning)
            n, mm, m, v, sk, kurt = stats.describe(xp.asarray(4.)[()])
        assert n == 1
        xp_assert_equal(mm[0], xp.asarray(4.0))
        xp_assert_equal(mm[1], xp.asarray(4.0))
        xp_assert_equal(m, xp.asarray(4.0))
        xp_assert_equal(v ,xp.asarray(xp.nan))
        xp_assert_equal(sk, xp.asarray(xp.nan))
        xp_assert_equal(kurt, xp.asarray(xp.nan))

    def test_describe_numbers(self, xp):
        x = xp.concat((xp.ones((3, 4)), xp.full((2, 4), 2.)))
        nc = 5
        mmc = (xp.asarray([1., 1., 1., 1.]), xp.asarray([2., 2., 2., 2.]))
        mc = xp.asarray([1.4, 1.4, 1.4, 1.4])
        vc = xp.asarray([0.3, 0.3, 0.3, 0.3])
        skc = xp.asarray([0.40824829046386357] * 4)
        kurtc = xp.asarray([-1.833333333333333] * 4)
        n, mm, m, v, sk, kurt = stats.describe(x)
        assert n == nc
        xp_assert_equal(mm[0], mmc[0])
        xp_assert_equal(mm[1], mmc[1])
        xp_assert_close(m, mc, rtol=4 * xp.finfo(m.dtype).eps)
        xp_assert_close(v, vc, rtol=4 * xp.finfo(m.dtype).eps)
        xp_assert_close(sk, skc)
        xp_assert_close(kurt, kurtc)

        n, mm, m, v, sk, kurt = stats.describe(x.T, axis=1)
        assert n == nc
        xp_assert_equal(mm[0], mmc[0])
        xp_assert_equal(mm[1], mmc[1])
        xp_assert_close(m, mc, rtol=4 * xp.finfo(m.dtype).eps)
        xp_assert_close(v, vc, rtol=4 * xp.finfo(m.dtype).eps)
        xp_assert_close(sk, skc)
        xp_assert_close(kurt, kurtc)

    def describe_nan_policy_omit_test(self):
        x = np.arange(10.)
        x[9] = np.nan

        nc, mmc = (9, (0.0, 8.0))
        mc = 4.0
        vc = 7.5
        skc = 0.0
        kurtc = -1.2300000000000002
        n, mm, m, v, sk, kurt = stats.describe(x, nan_policy='omit')
        assert_equal(n, nc)
        assert_equal(mm, mmc)
        assert_equal(m, mc)
        assert_equal(v, vc)
        assert_array_almost_equal(sk, skc)
        assert_array_almost_equal(kurt, kurtc, decimal=13)

    def test_describe_nan_policy_other(self, xp):
        x = xp.arange(10.)
        x = xp.where(x==9, xp.nan, x)

        if is_lazy_array(x):
            with pytest.raises(TypeError, match='not supported for lazy arrays'):
                stats.describe(x, nan_policy='raise')
        else:
            with pytest.raises(ValueError, match='The input contains nan values'):
                stats.describe(x, nan_policy='raise')

        n, mm, m, v, sk, kurt = stats.describe(x, nan_policy='propagate')
        ref = xp.asarray(xp.nan)[()]
        assert n == 10
        xp_assert_equal(mm[0], ref)
        xp_assert_equal(mm[1], ref)
        xp_assert_equal(m, ref)
        xp_assert_equal(v, ref)
        xp_assert_equal(sk, ref)
        xp_assert_equal(kurt, ref)

        if is_numpy(xp):
            self.describe_nan_policy_omit_test()
        elif is_lazy_array(x):
            with pytest.raises(TypeError, match='not supported for lazy arrays'):
                stats.describe(x, nan_policy='omit')

        message = 'nan_policy must be one of...'
        with pytest.raises(ValueError, match=message):
            stats.describe(x, nan_policy='foobar')

    def test_describe_result_attributes(self):
        # some result attributes are tuples, which aren't meant to be compared
        # with `xp_assert_close`
        actual = stats.describe(np.arange(5.))
        attributes = ('nobs', 'minmax', 'mean', 'variance', 'skewness', 'kurtosis')
        check_named_results(actual, attributes)

    def test_describe_ddof(self, xp):
        x = xp.concat((xp.ones((3, 4)), xp.full((2, 4), 2.)))
        nc = 5
        mmc = (xp.asarray([1., 1., 1., 1.]), xp.asarray([2., 2., 2., 2.]))
        mc = xp.asarray([1.4, 1.4, 1.4, 1.4])
        vc = xp.asarray([0.24, 0.24, 0.24, 0.24])
        skc = xp.asarray([0.40824829046386357] * 4)
        kurtc = xp.asarray([-1.833333333333333] * 4)
        n, mm, m, v, sk, kurt = stats.describe(x, ddof=0)
        assert n == nc
        xp_assert_equal(mm[0], mmc[0])
        xp_assert_equal(mm[1], mmc[1])
        xp_assert_close(m, mc)
        xp_assert_close(v, vc)
        xp_assert_close(sk, skc)
        xp_assert_close(kurt, kurtc)

    def test_describe_axis_none(self, xp):
        x = xp.concat((xp.ones((3, 4)), xp.full((2, 4), 2.)))

        # expected values
        nc = 20
        mmc = (xp.asarray(1.0), xp.asarray(2.0))
        mc = xp.asarray(1.3999999999999999)
        vc = xp.asarray(0.25263157894736848)
        skc = xp.asarray(0.4082482904638634)
        kurtc = xp.asarray(-1.8333333333333333)

        # actual values
        n, mm, m, v, sk, kurt = stats.describe(x, axis=None)

        assert n == nc
        xp_assert_equal(mm[0], mmc[0])
        xp_assert_equal(mm[1], mmc[1])
        xp_assert_close(m, mc)
        xp_assert_close(v, vc)
        xp_assert_close(sk, skc)
        xp_assert_close(kurt, kurtc)

    def test_describe_empty(self, xp):
        message = "The input must not be empty."
        with pytest.raises(ValueError, match=message):
            stats.describe(xp.asarray([]))


class NormalityTests:

    @pytest.mark.parametrize("alternative", ['two-sided', 'less', 'greater'])
    def test_against_R(self, alternative, xp):
        # testa against R `dagoTest` from package `fBasics`
        # library(fBasics)
        # options(digits=16)
        # x = c(-2, -1, 0, 1, 2, 3)**2
        # x = rep(x, times=4)
        # test_result <- dagoTest(x)
        # test_result@test$statistic
        # test_result@test$p.value
        test_name = self.test_name
        test_fun = getattr(stats, test_name)
        ref_statistic= xp.asarray(self.case_ref[0])
        ref_pvalue = xp.asarray(self.case_ref[1])

        kwargs = {}
        if alternative in {'less', 'greater'}:
            if test_name in {'skewtest', 'kurtosistest'}:
                ref_pvalue = ref_pvalue/2 if alternative == "less" else 1-ref_pvalue/2
                ref_pvalue = 1-ref_pvalue if test_name == 'skewtest' else ref_pvalue
                kwargs['alternative'] = alternative
            else:
                pytest.skip('`alternative` not available for `normaltest`')

        x = xp.asarray((-2, -1, 0, 1, 2, 3.)*4)**2
        res = test_fun(x, **kwargs)
        res_statistic, res_pvalue = res
        xp_assert_close(res_statistic, ref_statistic)
        xp_assert_close(res_pvalue, ref_pvalue)
        check_named_results(res, ('statistic', 'pvalue'), xp=xp)

    def test_nan(self, xp):
        # nan in input -> nan output (default nan_policy='propagate')
        test_fun = getattr(stats, self.test_name)
        x = xp.arange(30.)
        NaN = xp.asarray(xp.nan, dtype=x.dtype)
        x = xp.where(x == 29, NaN, x)
        with np.errstate(invalid="ignore"):
            res = test_fun(x)
            xp_assert_equal(res.statistic, NaN)
            xp_assert_equal(res.pvalue, NaN)


@make_xp_test_case(stats.skewtest)
class TestSkewTest(NormalityTests):
    test_name = 'skewtest'
    case_ref = (1.98078826090875881, 0.04761502382843208)  # statistic, pvalue

    def test_intuitive(self, xp):
        # intuitive tests; see gh-13549. skewnorm with parameter 1 has skew > 0
        a1 = stats.skewnorm.rvs(a=1, size=10000, random_state=123)
        a1_xp = xp.asarray(a1)
        pval = stats.skewtest(a1_xp, alternative='greater').pvalue
        xp_assert_close(pval, xp.asarray(0.0, dtype=a1_xp.dtype), atol=9e-6)

    def test_skewtest_too_few_observations(self, xp):
        # Regression test for ticket #1492.
        # skewtest requires at least 8 observations; 7 should warn and return NaN.
        stats.skewtest(xp.arange(8.0))

        x = xp.arange(7.0)

        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            res = stats.skewtest(x)
            NaN = xp.asarray(xp.nan)
            xp_assert_equal(res.statistic, NaN)
            xp_assert_equal(res.pvalue, NaN)


@make_xp_test_case(stats.kurtosistest)
class TestKurtosisTest(NormalityTests):
    test_name = 'kurtosistest'
    case_ref = (-0.01403734404759738, 0.98880018772590561)  # statistic, pvalue

    def test_intuitive(self, xp):
        # intuitive tests; see gh-13549. excess kurtosis of laplace is 3 > 0
        a2 = stats.laplace.rvs(size=10000, random_state=123)
        a2_xp = xp.asarray(a2)
        pval = stats.kurtosistest(a2_xp, alternative='greater').pvalue
        xp_assert_close(pval, xp.asarray(0.0, dtype=a2_xp.dtype), atol=1e-15)

    def test_gh9033_regression(self, xp):
        # regression test for issue gh-9033: x clearly non-normal but power of
        # negative denom needs to be handled correctly to reject normality
        counts = [128, 0, 58, 7, 0, 41, 16, 0, 0, 167]
        x = np.hstack([np.full(c, i) for i, c in enumerate(counts)])
        x = xp.asarray(x, dtype=xp.float64)
        assert stats.kurtosistest(x)[1] < 0.01

    def test_kurtosistest_too_few_observations(self, xp):
        # kurtosistest requires at least 5 observations; 4 should warn and return NaN.
        # Regression test for ticket #1425.
        stats.kurtosistest(xp.arange(5.0))

        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            res = stats.kurtosistest(xp.arange(4.))
            NaN = xp.asarray(xp.nan)
            xp_assert_equal(res.statistic, NaN)
            xp_assert_equal(res.pvalue, NaN)


@make_xp_test_case(stats.normaltest)
class TestNormalTest(NormalityTests):
    test_name = 'normaltest'
    case_ref = (3.92371918158185551, 0.14059672529747502)  # statistic, pvalue

    def test_too_few_observations(self, xp):
        stats.normaltest(xp.arange(8.))

        # 1D sample has too few observations -> warning / NaN output
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            res = stats.normaltest(xp.arange(7.))
            NaN = xp.asarray(xp.nan)
            xp_assert_equal(res.statistic, NaN)
            xp_assert_equal(res.pvalue, NaN)


class TestRankSums:

    rng = np.random.default_rng(3417115752)
    x, y = rng.random((2, 10))

    @pytest.mark.parametrize('alternative', ['less', 'greater', 'two-sided'])
    def test_ranksums_result_attributes(self, alternative):
        # ranksums pval = mannwhitneyu pval w/out continuity or tie correction
        res1 = stats.ranksums(self.x, self.y,
                              alternative=alternative).pvalue
        res2 = stats.mannwhitneyu(self.x, self.y, use_continuity=False,
                                  alternative=alternative).pvalue
        assert_allclose(res1, res2)

    def test_ranksums_named_results(self):
        res = stats.ranksums(self.x, self.y)
        check_named_results(res, ('statistic', 'pvalue'))

    def test_input_validation(self):
        with assert_raises(ValueError, match="`alternative` must be 'less'"):
            stats.ranksums(self.x, self.y, alternative='foobar')


@make_xp_test_case(stats.jarque_bera)
class TestJarqueBera:
    def test_jarque_bera_against_R(self, xp):
        # library(tseries)
        # options(digits=16)
        # x < - rnorm(5)
        # jarque.bera.test(x)
        x = [-0.160104223201523288,  1.131262000934478040, -0.001235254523709458,
             -0.776440091309490987, -2.072959999533182884]
        x = xp.asarray(x)
        ref = xp.asarray([0.17651605223752, 0.9155246169805])
        res = stats.jarque_bera(x)
        xp_assert_close(res.statistic, ref[0])
        xp_assert_close(res.pvalue, ref[1])

    @skip_xp_backends(np_only=True)
    def test_jarque_bera_array_like(self, xp):
        # array-like only relevant for NumPy
        rng = np.random.default_rng(9294968266)
        x = rng.standard_normal(size=100000)

        jb_test1 = JB1, p1 = stats.jarque_bera(list(x))
        jb_test2 = JB2, p2 = stats.jarque_bera(tuple(x))
        jb_test3 = JB3, p3 = stats.jarque_bera(x.reshape(2, 50000))

        assert JB1 == JB2 == JB3 == jb_test1.statistic == jb_test2.statistic == jb_test3.statistic  # noqa: E501
        assert p1 == p2 == p3 == jb_test1.pvalue == jb_test2.pvalue == jb_test3.pvalue

    @skip_xp_backends('array_api_strict', reason='Noisy; see TestSkew')
    def test_jarque_bera_too_few_observations(self, xp):
        x = xp.asarray([])

        with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
            res = stats.jarque_bera(x)
            NaN = xp.asarray(xp.nan)
            xp_assert_equal(res.statistic, NaN)
            xp_assert_equal(res.pvalue, NaN)

    def test_axis(self, xp):
        rng = np.random.RandomState(seed=122398129)
        x = xp.asarray(rng.random(size=(2, 45)))

        res = stats.jarque_bera(x, axis=None)
        ref = stats.jarque_bera(xp.reshape(x, (-1,)))
        xp_assert_equal(res.statistic, ref.statistic)
        xp_assert_equal(res.pvalue, ref.pvalue)

        res = stats.jarque_bera(x, axis=1)
        s0, p0 = stats.jarque_bera(x[0, :])
        s1, p1 = stats.jarque_bera(x[1, :])
        xp_assert_close(res.statistic, xp.stack([s0, s1]))
        xp_assert_close(res.pvalue, xp.stack([p0, p1]))

        resT = stats.jarque_bera(x.T, axis=0)
        xp_assert_close(res.statistic, resT.statistic)
        xp_assert_close(res.pvalue, resT.pvalue)


def test_pointbiserial():
    # same as mstats test except for the nan
    # Test data: https://web.archive.org/web/20060504220742/https://support.sas.com/ctx/samples/index.jsp?sid=490&tab=output
    x = [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,0,
         0,0,0,0,1]
    y = [14.8,13.8,12.4,10.1,7.1,6.1,5.8,4.6,4.3,3.5,3.3,3.2,3.0,
         2.8,2.8,2.5,2.4,2.3,2.1,1.7,1.7,1.5,1.3,1.3,1.2,1.2,1.1,
         0.8,0.7,0.6,0.5,0.2,0.2,0.1]
    assert_almost_equal(stats.pointbiserialr(x, y)[0], 0.36149, 5)

    # test for namedtuple attribute results
    attributes = ('correlation', 'pvalue')
    res = stats.pointbiserialr(x, y)
    check_named_results(res, attributes)
    assert_equal(res.correlation, res.statistic)


def test_obrientransform():
    # A couple tests calculated by hand.
    x1 = np.array([0, 2, 4])
    t1 = stats.obrientransform(x1)
    expected = [7, -2, 7]
    assert_allclose(t1[0], expected)

    x2 = np.array([0, 3, 6, 9])
    t2 = stats.obrientransform(x2)
    expected = np.array([30, 0, 0, 30])
    assert_allclose(t2[0], expected)

    # Test two arguments.
    a, b = stats.obrientransform(x1, x2)
    assert_equal(a, t1[0])
    assert_equal(b, t2[0])

    # Test three arguments.
    a, b, c = stats.obrientransform(x1, x2, x1)
    assert_equal(a, t1[0])
    assert_equal(b, t2[0])
    assert_equal(c, t1[0])

    # This is a regression test to check np.var replacement.
    # The author of this test didn't separately verify the numbers.
    x1 = np.arange(5)
    result = np.array(
      [[5.41666667, 1.04166667, -0.41666667, 1.04166667, 5.41666667],
       [21.66666667, 4.16666667, -1.66666667, 4.16666667, 21.66666667]])
    assert_array_almost_equal(stats.obrientransform(x1, 2*x1), result, decimal=8)

    # Example from "O'Brien Test for Homogeneity of Variance"
    # by Herve Abdi.
    values = range(5, 11)
    reps = np.array([5, 11, 9, 3, 2, 2])
    data = np.repeat(values, reps)
    transformed_values = np.array([3.1828, 0.5591, 0.0344,
                                   1.6086, 5.2817, 11.0538])
    expected = np.repeat(transformed_values, reps)
    result = stats.obrientransform(data)
    assert_array_almost_equal(result[0], expected, decimal=4)


def check_equal_xmean(*args, xp, mean_fun, axis=None, dtype=None,
                      rtol=1e-7, weights=None):
    # Note this doesn't test when axis is not specified
    dtype = dtype or xp.float64
    if len(args) == 2:
        array_like, desired = args
    else:
        array_like, p, desired = args
    array_like = xp.asarray(array_like, dtype=dtype)
    desired = xp.asarray(desired, dtype=dtype)
    weights = xp.asarray(weights, dtype=dtype) if weights is not None else weights
    args = (array_like,) if len(args) == 2 else (array_like, p)
    x = mean_fun(*args, axis=axis, dtype=dtype, weights=weights)
    xp_assert_close(x, desired, rtol=rtol)


def check_equal_gmean(*args, **kwargs):
    return check_equal_xmean(*args, mean_fun=stats.gmean, **kwargs)


def check_equal_hmean(*args, **kwargs):
    return check_equal_xmean(*args, mean_fun=stats.hmean, **kwargs)


def check_equal_pmean(*args, **kwargs):
    return check_equal_xmean(*args, mean_fun=stats.pmean, **kwargs)


@make_xp_test_case(stats.hmean)
class TestHMean:
    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_0(self, xp):
        a = [1, 0, 2]
        desired = 0
        check_equal_hmean(a, desired, xp=xp)

    def test_1d(self, xp):
        #  Test a 1d case
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
        desired = 34.1417152147
        check_equal_hmean(a, desired, xp=xp)

        a = [1, 2, 3, 4]
        desired = 4. / (1. / 1 + 1. / 2 + 1. / 3 + 1. / 4)
        check_equal_hmean(a, desired, xp=xp)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_1d_with_zero(self, xp):
        a = np.array([1, 0])
        desired = 0.0
        check_equal_hmean(a, desired, xp=xp, rtol=0.0)

    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered:RuntimeWarning"
    ) # for dask
    def test_1d_with_negative_value(self, xp):
        a = np.array([1, 0, -1])
        message = "The harmonic mean is only defined..."
        with pytest.warns(RuntimeWarning, match=message):
            check_equal_hmean(a, xp.nan, xp=xp, rtol=0.0)

    # Note the next tests use axis=None as default, not axis=0
    def test_2d(self, xp):
        #  Test a 2d case
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = 38.6696271841
        check_equal_hmean(np.array(a), desired, xp=xp)

    def test_2d_axis0(self, xp):
        #  Test a 2d case with axis=0
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = np.array([22.88135593, 39.13043478, 52.90076336, 65.45454545])
        check_equal_hmean(a, desired, axis=0, xp=xp)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_2d_axis0_with_zero(self, xp):
        a = [[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = np.array([22.88135593, 0.0, 52.90076336, 65.45454545])
        check_equal_hmean(a, desired, axis=0, xp=xp)

    def test_2d_axis1(self, xp):
        #  Test a 2d case with axis=1
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = np.array([19.2, 63.03939962, 103.80078637])
        check_equal_hmean(a, desired, axis=1, xp=xp)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_2d_axis1_with_zero(self, xp):
        a = [[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = np.array([0.0, 63.03939962, 103.80078637])
        check_equal_hmean(a, desired, axis=1, xp=xp)

    @skip_xp_backends(
        np_only=True,
        reason='array-likes only supported for NumPy backend',
    )
    def test_weights_1d_list(self, xp):
        # Desired result from:
        # https://www.hackmath.net/en/math-problem/35871
        a = [2, 10, 6]
        weights = [10, 5, 3]
        desired = 3.
        # all the other tests use `check_equal_hmean`, which now converts
        # the input to an xp-array before calling `hmean`. This time, check
        # that the function still accepts the lists of ints.
        res = stats.hmean(a, weights=weights)
        xp_assert_close(res, np.asarray(desired), rtol=1e-5)

    def test_weights_1d(self, xp):
        # Desired result from:
        # https://www.hackmath.net/en/math-problem/35871
        a = np.asarray([2, 10, 6])
        weights = np.asarray([10, 5, 3])
        desired = 3
        check_equal_hmean(a, desired, weights=weights, rtol=1e-5, xp=xp)

    def test_weights_2d_axis0(self, xp):
        # Desired result from:
        # https://www.hackmath.net/en/math-problem/35871
        a = np.array([[2, 5], [10, 5], [6, 5]])
        weights = np.array([[10, 1], [5, 1], [3, 1]])
        desired = np.array([3, 5])
        check_equal_hmean(a, desired, axis=0, weights=weights, rtol=1e-5, xp=xp)

    def test_weights_2d_axis1(self, xp):
        # Desired result from:
        # https://www.hackmath.net/en/math-problem/35871
        a = np.array([[2, 10, 6], [7, 7, 7]])
        weights = np.array([[10, 5, 3], [1, 1, 1]])
        desired = np.array([3, 7])
        check_equal_hmean(a, desired, axis=1, weights=weights, rtol=1e-5, xp=xp)

    @skip_xp_invalid_arg
    def test_weights_masked_1d_array(self, xp):
        # Desired result from:
        # https://www.hackmath.net/en/math-problem/35871
        a = np.array([2, 10, 6, 42])
        weights = np.ma.array([10, 5, 3, 42], mask=[0, 0, 0, 1])
        desired = 3
        xp = np.ma  # check_equal_hmean uses xp.asarray; this will preserve the mask
        check_equal_hmean(a, desired, weights=weights, rtol=1e-5,
                          dtype=np.float64, xp=xp)


@make_xp_test_case(stats.gmean)
class TestGMean:
    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered in log:RuntimeWarning:dask"
    )
    def test_0(self, xp):
        a = [1, 0, 2]
        desired = 0
        check_equal_gmean(a, desired, xp=xp)

    def test_1d(self, xp):
        #  Test a 1d case
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
        desired = 45.2872868812
        check_equal_gmean(a, desired, xp=xp)

        a = [1, 2, 3, 4]
        desired = power(1 * 2 * 3 * 4, 1. / 4.)
        check_equal_gmean(a, desired, rtol=1e-14, xp=xp)

        a = array([1, 2, 3, 4], float32)
        desired = power(1 * 2 * 3 * 4, 1. / 4.)
        check_equal_gmean(a, desired, dtype=xp.float32, xp=xp)

    # Note the next tests use axis=None as default, not axis=0
    def test_2d(self, xp):
        #  Test a 2d case
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = 52.8885199
        check_equal_gmean(a, desired, xp=xp)

    def test_2d_axis0(self, xp):
        #  Test a 2d case with axis=0
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = np.array([35.56893304, 49.32424149, 61.3579244, 72.68482371])
        check_equal_gmean(a, desired, axis=0, xp=xp)

        a = array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]])
        desired = array([1, 2, 3, 4])
        check_equal_gmean(a, desired, axis=0, rtol=1e-14, xp=xp)

    def test_2d_axis1(self, xp):
        #  Test a 2d case with axis=1
        a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
        desired = np.array([22.13363839, 64.02171746, 104.40086817])
        check_equal_gmean(a, desired, axis=1, xp=xp)

        a = array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]])
        v = power(1 * 2 * 3 * 4, 1. / 4.)
        desired = array([v, v, v])
        check_equal_gmean(a, desired, axis=1, rtol=1e-14, xp=xp)

    def test_large_values(self, xp):
        a = array([1e100, 1e200, 1e300])
        desired = 1e200
        check_equal_gmean(a, desired, rtol=1e-13, xp=xp)

    @pytest.mark.filterwarnings(
        "ignore:divide by zero encountered in log:RuntimeWarning:dask"
    )
    def test_1d_with_0(self, xp):
        #  Test a 1d case with zero element
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 0]
        desired = 0.0  # due to exp(-inf)=0
        with np.errstate(all='ignore'):
            check_equal_gmean(a, desired, xp=xp)

    @pytest.mark.filterwarnings(
        "ignore:invalid value encountered in log:RuntimeWarning:dask"
    )
    def test_1d_neg(self, xp):
        #  Test a 1d case with negative element
        a = [10, 20, 30, 40, 50, 60, 70, 80, 90, -1]
        desired = np.nan  # due to log(-1) = nan
        with np.errstate(invalid='ignore'):
            check_equal_gmean(a, desired, xp=xp)

    @skip_xp_backends(
        np_only=True,
        reason='array-likes only supported for NumPy backend',
    )
    def test_weights_1d_list(self, xp):
        # Desired result from:
        # https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
        a = [1, 2, 3, 4, 5]
        weights = [2, 5, 6, 4, 3]
        desired = 2.77748

        # all the other tests use `check_equal_gmean`, which now converts
        # the input to an xp-array before calling `gmean`. This time, check
        # that the function still accepts the lists of ints.
        res = stats.gmean(a, weights=weights)
        xp_assert_close(res, np.asarray(desired), rtol=1e-5)

    def test_weights_1d(self, xp):
        # Desired result from:
        # https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
        a = np.array([1, 2, 3, 4, 5])
        weights = np.array([2, 5, 6, 4, 3])
        desired = 2.77748
        check_equal_gmean(a, desired, weights=weights, rtol=1e-5, xp=xp)

    @skip_xp_invalid_arg
    def test_weights_masked_1d_array(self, xp):
        # Desired result from:
        # https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
        a = np.array([1, 2, 3, 4, 5, 6])
        weights = np.ma.array([2, 5, 6, 4, 3, 5], mask=[0, 0, 0, 0, 0, 1])
        desired = 2.77748
        xp = np.ma  # check_equal_gmean uses xp.asarray; this will preserve the mask
        check_equal_gmean(a, desired, weights=weights, rtol=1e-5,
                          dtype=np.float64, xp=xp)


@make_xp_test_case(stats.pmean)
class TestPMean:

    def pmean_reference(a, p):
        return (np.sum(a**p) / a.size)**(1/p)

    def wpmean_reference(a, p, weights):
        return (np.sum(weights * a**p) / np.sum(weights))**(1/p)

    def test_bad_exponent(self, xp):
        with pytest.raises(ValueError, match='Power mean only defined for'):
            stats.pmean(xp.asarray([1, 2, 3]), xp.asarray([0]))

    def test_1d(self, xp):
        a, p = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], 3.5
        desired = TestPMean.pmean_reference(np.array(a), p)
        check_equal_pmean(a, p, desired, xp=xp)

        a, p = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], -2.5
        desired = TestPMean.pmean_reference(np.array(a), p)
        check_equal_pmean(a, p, desired, xp=xp)

        a, p = [1, 2, 3, 4], 2
        desired = np.sqrt((1**2 + 2**2 + 3**2 + 4**2) / 4)
        check_equal_pmean(a, p, desired, xp=xp)

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    def test_1d_with_zero(self, xp):
        a, p = np.array([1, 0]), -1
        desired = 0.0
        check_equal_pmean(a, p, desired, rtol=0.0, xp=xp)

    def test_1d_with_negative_value(self, xp):
        a, p = np.array([1, 0, -1]), 1.23
        message = "The power mean is only defined..."
        with pytest.warns(RuntimeWarning, match=message):
            check_equal_pmean(a, p, xp.nan, xp=xp)

    @pytest.mark.parametrize(
        ("a", "p"),
        [([[10, 20], [50, 60], [90, 100]], -0.5),
         (np.array([[10, 20], [50, 60], [90, 100]]), 0.5)]
    )
    def test_2d_axisnone(self, a, p, xp):
        desired = TestPMean.pmean_reference(np.array(a), p)
        check_equal_pmean(a, p, desired, xp=xp)

    @pytest.mark.parametrize(
        ("a", "p"),
        [([[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], -0.5),
         ([[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], 0.5)]
    )
    def test_2d_axis0(self, a, p, xp):
        desired = [
            TestPMean.pmean_reference(
                np.array([a[i][j] for i in range(len(a))]), p
            )
            for j in range(len(a[0]))
        ]
        check_equal_pmean(a, p, desired, axis=0, xp=xp)

    @pytest.mark.parametrize(
        ("a", "p"),
        [([[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], -0.5),
         ([[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], 0.5)]
    )
    def test_2d_axis1(self, a, p, xp):
        desired = [TestPMean.pmean_reference(np.array(a_), p) for a_ in a]
        check_equal_pmean(a, p, desired, axis=1, xp=xp)

    def test_weights_1d(self, xp):
        a, p = [2, 10, 6], -1.23456789
        weights = [10, 5, 3]
        desired = TestPMean.wpmean_reference(np.array(a), p, weights)
        check_equal_pmean(a, p, desired, weights=weights, rtol=1e-5, xp=xp)

    @skip_xp_backends(
        np_only=True,
        reason='array-likes only supported for NumPy backend',
    )
    def test_weights_1d_list(self, xp):
        a, p = [2, 10, 6], -1.23456789
        weights = [10, 5, 3]
        desired = TestPMean.wpmean_reference(np.array(a), p, weights)
        # all the other tests use `check_equal_pmean`, which now converts
        # the input to an xp-array before calling `pmean`. This time, check
        # that the function still accepts the lists of ints.
        res = stats.pmean(a, p, weights=weights)
        xp_assert_close(res, np.asarray(desired), rtol=1e-5)

    @skip_xp_invalid_arg
    def test_weights_masked_1d_array(self, xp):
        a, p = np.array([2, 10, 6, 42]), 1
        weights = np.ma.array([10, 5, 3, 42], mask=[0, 0, 0, 1])
        desired = np.average(a, weights=weights)
        xp = np.ma  # check_equal_pmean uses xp.asarray; this will preserve the mask
        check_equal_pmean(a, p, desired, weights=weights, rtol=1e-5,
                          dtype=np.float64, xp=xp)

    @pytest.mark.parametrize(
        ("axis", "fun_name", "p"),
        [(None, "wpmean_reference", 9.87654321),
         (0, "gmean", 0),
         (1, "hmean", -1)]
    )
    def test_weights_2d(self, axis, fun_name, p, xp):
        if fun_name == 'wpmean_reference':
            def fun(a, axis, weights):
                return TestPMean.wpmean_reference(a, p, weights)
        else:
            fun = getattr(stats, fun_name)
        a = np.array([[2, 5], [10, 5], [6, 5]])
        weights = np.array([[10, 1], [5, 1], [3, 1]])
        desired = fun(a, axis=axis, weights=weights)
        check_equal_pmean(a, p, desired, axis=axis, weights=weights, rtol=1e-5, xp=xp)

    def test_infinite_p_gh23111(self):
        # gh-23111 reported that `pmean` didn't work properly with infinite `p`;
        # check that this raises an appropriate error message
        message = "Power mean only implemented for finite `p`"
        with pytest.raises(NotImplementedError, match=message):
            stats.pmean([2], np.inf)


@make_xp_test_case(stats.gstd)
class TestGSTD:
    # must add 1 as `gstd` is only defined for positive values
    array_1d = (np.arange(2 * 3 * 4) + 1).tolist()
    gstd_array_1d = 2.294407613602
    array_3d = np.reshape(array_1d, (2, 3, 4)).tolist()

    def test_1d_array(self, xp):
        gstd_actual = stats.gstd(xp.asarray(self.array_1d))
        xp_assert_close(gstd_actual, xp.asarray(self.gstd_array_1d))

    @skip_xp_backends(np_only=True, reason="Only NumPy supports array-like input")
    def test_1d_numeric_array_like_input(self, xp):
        gstd_actual = stats.gstd(tuple(self.array_1d))
        assert_allclose(gstd_actual, self.gstd_array_1d)

    @skip_xp_invalid_arg
    def test_raises_error_non_numeric_input(self, xp):
        message = "could not convert string to float|The DType..."
        with pytest.raises((ValueError, TypeError), match=message):
            stats.gstd('You cannot take the logarithm of a string.')

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    @pytest.mark.parametrize('bad_value', (0, -1, np.inf, np.nan))
    def test_returns_nan_invalid_value(self, bad_value, xp):
        x = xp.asarray(self.array_1d + [bad_value])
        if np.isfinite(bad_value) and not is_lazy_array(x):
            message = "The geometric standard deviation is only defined..."
            with pytest.warns(RuntimeWarning, match=message):
                res = stats.gstd(x)
        else:
            res = stats.gstd(x)
        xp_assert_equal(res, xp.asarray(np.nan))

    def test_propagates_nan_values(self, xp):
        a = xp.asarray([[1, 1, 1, 16], [xp.nan, 1, 2, 3]])
        gstd_actual = stats.gstd(a, axis=1)
        xp_assert_close(gstd_actual, xp.asarray([4, np.nan]))

    @xfail_xp_backends("jax.numpy", reason="returns subnormal instead of nan")
    def test_ddof_equal_to_number_of_observations(self, xp):
        x = xp.asarray(self.array_1d)
        res = stats.gstd(x, ddof=x.shape[0])
        xp_assert_equal(res, xp.asarray(xp.nan))

    def test_3d_array(self, xp):
        x = xp.asarray(self.array_3d)
        gstd_actual = stats.gstd(x, axis=None)
        ref = xp.asarray(self.gstd_array_1d)
        xp_assert_close(gstd_actual, ref)

    def test_3d_array_axis_type_tuple(self, xp):
        x = xp.asarray(self.array_3d)
        gstd_actual = stats.gstd(x, axis=(1, 2))
        ref = xp.asarray([2.12939215, 1.22120169])
        xp_assert_close(gstd_actual, ref)

    def test_3d_array_axis_0(self, xp):
        x = xp.asarray(self.array_3d)
        gstd_actual = stats.gstd(x, axis=0)
        gstd_desired = xp.asarray([
            [6.1330555493918, 3.958900210120, 3.1206598248344, 2.6651441426902],
            [2.3758135028411, 2.174581428192, 2.0260062829505, 1.9115518327308],
            [1.8205343606803, 1.746342404566, 1.6846557065742, 1.6325269194382]
        ])
        xp_assert_close(gstd_actual, gstd_desired)

    def test_3d_array_axis_1(self, xp):
        x = xp.asarray(self.array_3d)
        gstd_actual = stats.gstd(x, axis=1)
        gstd_desired = xp.asarray([
            [3.118993630946, 2.275985934063, 1.933995977619, 1.742896469724],
            [1.271693593916, 1.254158641801, 1.238774141609, 1.225164057869]
        ])
        xp_assert_close(gstd_actual, gstd_desired)

    def test_3d_array_axis_2(self, xp):
        x = xp.asarray(self.array_3d)
        gstd_actual = stats.gstd(x, axis=2)
        gstd_desired = xp.asarray([
            [1.8242475707664, 1.2243686572447, 1.1318311657788],
            [1.0934830582351, 1.0724479791887, 1.0591498540749]
        ])
        xp_assert_close(gstd_actual, gstd_desired)


def test_binomtest():
    # precision tests compared to R for ticket:986
    pp = np.concatenate((np.linspace(0.1, 0.2, 5),
                         np.linspace(0.45, 0.65, 5),
                         np.linspace(0.85, 0.95, 5)))
    n = 501
    x = 450
    results = [0.0, 0.0, 1.0159969301994141e-304,
               2.9752418572150531e-275, 7.7668382922535275e-250,
               2.3381250925167094e-099, 7.8284591587323951e-081,
               9.9155947819961383e-065, 2.8729390725176308e-050,
               1.7175066298388421e-037, 0.0021070691951093692,
               0.12044570587262322, 0.88154763174802508, 0.027120993063129286,
               2.6102587134694721e-006]

    for p, res in zip(pp, results):
        assert_approx_equal(stats.binomtest(x, n, p).pvalue, res,
                            significant=12, err_msg=f'fail forp={p}')
    assert_approx_equal(stats.binomtest(50, 100, 0.1).pvalue,
                        5.8320387857343647e-024,
                        significant=12)


def test_binomtest2():
    # test added for issue #2384
    res2 = [
        [1.0, 1.0],
        [0.5, 1.0, 0.5],
        [0.25, 1.00, 1.00, 0.25],
        [0.125, 0.625, 1.000, 0.625, 0.125],
        [0.0625, 0.3750, 1.0000, 1.0000, 0.3750, 0.0625],
        [0.03125, 0.21875, 0.68750, 1.00000, 0.68750, 0.21875, 0.03125],
        [0.015625, 0.125000, 0.453125, 1.000000, 1.000000, 0.453125, 0.125000,
         0.015625],
        [0.0078125, 0.0703125, 0.2890625, 0.7265625, 1.0000000, 0.7265625,
         0.2890625, 0.0703125, 0.0078125],
        [0.00390625, 0.03906250, 0.17968750, 0.50781250, 1.00000000,
         1.00000000, 0.50781250, 0.17968750, 0.03906250, 0.00390625],
        [0.001953125, 0.021484375, 0.109375000, 0.343750000, 0.753906250,
         1.000000000, 0.753906250, 0.343750000, 0.109375000, 0.021484375,
         0.001953125]
    ]
    for k in range(1, 11):
        res1 = [stats.binomtest(v, k, 0.5).pvalue for v in range(k + 1)]
        assert_almost_equal(res1, res2[k-1], decimal=10)


def test_binomtest3():
    # test added for issue #2384
    # test when x == n*p and neighbors
    res3 = [stats.binomtest(v, v*k, 1./k).pvalue
            for v in range(1, 11) for k in range(2, 11)]
    assert_equal(res3, np.ones(len(res3), int))

    # > bt=c()
    # > for(i in as.single(1:10)) {
    # +     for(k in as.single(2:10)) {
    # +         bt = c(bt, binom.test(i-1, k*i,(1/k))$p.value);
    # +         print(c(i+1, k*i,(1/k)))
    # +     }
    # + }
    binom_testm1 = np.array([
         0.5, 0.5555555555555556, 0.578125, 0.5904000000000003,
         0.5981224279835393, 0.603430543396034, 0.607304096221924,
         0.610255656871054, 0.612579511000001, 0.625, 0.670781893004115,
         0.68853759765625, 0.6980101120000006, 0.703906431368616,
         0.70793209416498, 0.7108561134173507, 0.713076544331419,
         0.714820192935702, 0.6875, 0.7268709038256367, 0.7418963909149174,
         0.74986110468096, 0.7548015520398076, 0.7581671424768577,
         0.760607984787832, 0.762459425024199, 0.7639120677676575, 0.7265625,
         0.761553963657302, 0.774800934828818, 0.7818005980538996,
         0.78613491480358, 0.789084353140195, 0.7912217659828884,
         0.79284214559524, 0.794112956558801, 0.75390625, 0.7856929451142176,
         0.7976688481430754, 0.8039848974727624, 0.807891868948366,
         0.8105487660137676, 0.812473307174702, 0.8139318233591120,
         0.815075399104785, 0.7744140625, 0.8037322594985427,
         0.814742863657656, 0.8205425178645808, 0.8241275984172285,
         0.8265645374416, 0.8283292196088257, 0.829666291102775,
         0.8307144686362666, 0.7905273437499996, 0.8178712053954738,
         0.828116983756619, 0.833508948940494, 0.8368403871552892,
         0.839104213210105, 0.840743186196171, 0.84198481438049,
         0.8429580531563676, 0.803619384765625, 0.829338573944648,
         0.8389591907548646, 0.84401876783902, 0.84714369697889,
         0.8492667010581667, 0.850803474598719, 0.851967542858308,
         0.8528799045949524, 0.8145294189453126, 0.838881732845347,
         0.847979024541911, 0.852760894015685, 0.8557134656773457,
         0.8577190131799202, 0.85917058278431, 0.860270010472127,
         0.861131648404582, 0.823802947998047, 0.846984756807511,
         0.855635653643743, 0.860180994825685, 0.86298688573253,
         0.864892525675245, 0.866271647085603, 0.867316125625004,
         0.8681346531755114
        ])

    # > bt=c()
    # > for(i in as.single(1:10)) {
    # +     for(k in as.single(2:10)) {
    # +         bt = c(bt, binom.test(i+1, k*i,(1/k))$p.value);
    # +         print(c(i+1, k*i,(1/k)))
    # +     }
    # + }

    binom_testp1 = np.array([
         0.5, 0.259259259259259, 0.26171875, 0.26272, 0.2632244513031551,
         0.2635138663069203, 0.2636951804161073, 0.2638162407564354,
         0.2639010709000002, 0.625, 0.4074074074074074, 0.42156982421875,
         0.4295746560000003, 0.43473045988554, 0.4383309503172684,
         0.4409884859402103, 0.4430309389962837, 0.444649849401104, 0.6875,
         0.4927602499618962, 0.5096031427383425, 0.5189636628480,
         0.5249280070771274, 0.5290623300865124, 0.5320974248125793,
         0.5344204730474308, 0.536255847400756, 0.7265625, 0.5496019313526808,
         0.5669248746708034, 0.576436455045805, 0.5824538812831795,
         0.5866053321547824, 0.589642781414643, 0.5919618019300193,
         0.593790427805202, 0.75390625, 0.590868349763505, 0.607983393277209,
         0.617303847446822, 0.623172512167948, 0.627208862156123,
         0.6301556891501057, 0.632401894928977, 0.6341708982290303,
         0.7744140625, 0.622562037497196, 0.639236102912278, 0.648263335014579,
         0.65392850011132, 0.657816519817211, 0.660650782947676,
         0.662808780346311, 0.6645068560246006, 0.7905273437499996,
         0.6478843304312477, 0.6640468318879372, 0.6727589686071775,
         0.6782129857784873, 0.681950188903695, 0.684671508668418,
         0.686741824999918, 0.688369886732168, 0.803619384765625,
         0.668716055304315, 0.684360013879534, 0.6927642396829181,
         0.6980155964704895, 0.701609591890657, 0.7042244320992127,
         0.7062125081341817, 0.707775152962577, 0.8145294189453126,
         0.686243374488305, 0.7013873696358975, 0.709501223328243,
         0.714563595144314, 0.718024953392931, 0.7205416252126137,
         0.722454130389843, 0.723956813292035, 0.823802947998047,
         0.701255953767043, 0.715928221686075, 0.723772209289768,
         0.7286603031173616, 0.7319999279787631, 0.7344267920995765,
         0.736270323773157, 0.737718376096348
        ])

    res4_p1 = [stats.binomtest(v+1, v*k, 1./k).pvalue
               for v in range(1, 11) for k in range(2, 11)]
    res4_m1 = [stats.binomtest(v-1, v*k, 1./k).pvalue
               for v in range(1, 11) for k in range(2, 11)]

    assert_almost_equal(res4_p1, binom_testp1, decimal=13)
    assert_almost_equal(res4_m1, binom_testm1, decimal=13)


class TestTrim:
    # test trim functions
    def test_trim1(self):
        a = np.arange(11)
        assert_equal(np.sort(stats.trim1(a, 0.1)), np.arange(10))
        assert_equal(np.sort(stats.trim1(a, 0.2)), np.arange(9))
        assert_equal(np.sort(stats.trim1(a, 0.2, tail='left')),
                     np.arange(2, 11))
        assert_equal(np.sort(stats.trim1(a, 3/11., tail='left')),
                     np.arange(3, 11))
        assert_equal(stats.trim1(a, 1.0), [])
        assert_equal(stats.trim1(a, 1.0, tail='left'), [])

        # empty input
        assert_equal(stats.trim1([], 0.1), [])
        assert_equal(stats.trim1([], 3/11., tail='left'), [])
        assert_equal(stats.trim1([], 4/6.), [])

        # test axis
        a = np.arange(24).reshape(6, 4)
        ref = np.arange(4, 24).reshape(5, 4)  # first row trimmed

        axis = 0
        trimmed = stats.trim1(a, 0.2, tail='left', axis=axis)
        assert_equal(np.sort(trimmed, axis=axis), ref)

        axis = 1
        trimmed = stats.trim1(a.T, 0.2, tail='left', axis=axis)
        assert_equal(np.sort(trimmed, axis=axis), ref.T)

    def test_trimboth(self):
        a = np.arange(11)
        assert_equal(np.sort(stats.trimboth(a, 3/11.)), np.arange(3, 8))
        assert_equal(np.sort(stats.trimboth(a, 0.2)),
                     np.array([2, 3, 4, 5, 6, 7, 8]))
        assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(6, 4), 0.2)),
                     np.arange(4, 20).reshape(4, 4))
        assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(4, 6).T,
                                            2/6.)),
                     np.array([[2, 8, 14, 20], [3, 9, 15, 21]]))
        assert_raises(ValueError, stats.trimboth,
                      np.arange(24).reshape(4, 6).T, 4/6.)

        # empty input
        assert_equal(stats.trimboth([], 0.1), [])
        assert_equal(stats.trimboth([], 3/11.), [])
        assert_equal(stats.trimboth([], 4/6.), [])


@make_xp_test_case(stats.trim_mean)
class TestTrimMean:
    def test_trim_mean(self, xp):
        # don't use pre-sorted arrays
        idx = np.array([3, 5, 0, 1, 2, 4])
        a2 = np.arange(24).reshape(6, 4)[idx, :]
        a3 = np.arange(24).reshape(6, 4, order='F')[idx, :]

        xp_assert_equal(stats.trim_mean(xp.asarray(a3), 2/6.),
                        xp.asarray([2.5, 8.5, 14.5, 20.5]))
        xp_assert_equal(stats.trim_mean(xp.asarray(a2), 2/6.),
                        xp.asarray([10., 11., 12., 13.]))

        idx4 = np.array([1, 0, 3, 2])
        a4 = np.arange(24).reshape(4, 6)[idx4, :]
        xp_assert_equal(stats.trim_mean(xp.asarray(a4), 2/6.),
                        xp.asarray([9., 10., 11., 12., 13., 14.]))

        # shuffled arange(24) as array_like
        a = [7, 11, 12, 21, 16, 6, 22, 1, 5, 0, 18, 10, 17, 9, 19, 15, 23,
             20, 2, 14, 4, 13, 8, 3]
        xp_assert_equal(stats.trim_mean(xp.asarray(a), 2/6.), xp.asarray(11.5))
        xp_assert_equal(stats.trim_mean(xp.asarray([5, 4, 3, 1, 2, 0]), 2/6.),
                        xp.asarray(2.5))

        # check axis argument
        rng = np.random.default_rng(3417115752)
        a = rng.integers(20, size=(5, 6, 4, 7))
        a = xp.asarray(a)
        for axis in [0, 1, 2, 3, -1]:
            res1 = stats.trim_mean(a, 2/6., axis=axis)
            res2 = stats.trim_mean(xp.moveaxis(a, axis, 0), 2/6.)
            xp_assert_equal(res1, res2)

        res1 = stats.trim_mean(a, 2/6., axis=None)
        res2 = stats.trim_mean(xp_ravel(a), 2/6.)
        xp_assert_equal(res1, res2)

        with pytest.raises(ValueError, match="Proportion too big."):
            stats.trim_mean(a, 0.6)


    @pytest.mark.skip_xp_backends('jax.numpy', reason="lazy -> no _axis_nan_policy")
    @pytest.mark.skip_xp_backends('dask.array', reason="lazy -> no _axis_nan_policy")
    def test_empty_input(self, xp):
        # empty input
        with pytest.warns(SmallSampleWarning, match='too small'):
            xp_assert_equal(stats.trim_mean(xp.asarray([]), 0.0), xp.asarray(xp.nan))
        with pytest.warns(SmallSampleWarning, match='too small'):
            xp_assert_equal(stats.trim_mean(xp.asarray([]), 0.6), xp.asarray(xp.nan))


@make_xp_test_case(stats.sigmaclip)
class TestSigmaClip:
    def test_sigmaclip1(self, xp):
        a = xp.concat((xp.linspace(9.5, 10.5, 31),
                       xp.linspace(0., 20., 5)))
        fact = 4  # default
        c, low, upp = stats.sigmaclip(a)
        assert xp.min(c) > low
        assert xp.max(c) < upp
        xp_assert_equal(low, xp.mean(c) - fact*xp.std(c))
        xp_assert_equal(upp, xp.mean(c) + fact*xp.std(c))
        assert c.shape == a.shape

    def test_sigmaclip2(self, xp):
        a = xp.concat((xp.linspace(9.5, 10.5, 31),
                       xp.linspace(0., 20., 5)))
        fact = 1.5
        c, low, upp = stats.sigmaclip(a, fact, fact)
        assert xp.min(c) > low
        assert xp.max(c) < upp
        xp_assert_equal(low, xp.mean(c) - fact*xp.std(c))
        xp_assert_equal(upp, xp.mean(c) + fact*xp.std(c))
        assert c.shape == (4,)
        assert a.shape == (36,)

    def test_sigmaclip3(self, xp):
        a = xp.concat((xp.linspace(9.5, 10.5, 11),
                       xp.linspace(-100., -50., 3)))
        fact = 1.8
        c, low, upp = stats.sigmaclip(a, fact, fact)
        assert xp.min(c) > low
        assert xp.max(c) < upp
        xp_assert_equal(low, xp.mean(c) - fact*xp.std(c))
        xp_assert_equal(upp, xp.mean(c) + fact*xp.std(c))
        xp_assert_equal(c, xp.linspace(9.5, 10.5, 11))

    def test_sigmaclip_result_attributes(self, xp):
        a = xp.concat((xp.linspace(9.5, 10.5, 11),
                       xp.linspace(-100., -50., 3)))
        fact = 1.8
        res = stats.sigmaclip(a, fact, fact)
        attributes = ('clipped', 'lower', 'upper')
        check_named_results(res, attributes, xp=xp)

    def test_std_zero(self, xp):
        # regression test #8632
        x = xp.ones(10)
        xp_assert_equal(stats.sigmaclip(x)[0], x)


@make_xp_test_case(stats.alexandergovern)
class TestAlexanderGovern:
    def test_compare_dtypes(self):
        # leave this NumPy only
        args = [[13, 13, 13, 13, 13, 13, 13, 12, 12],
                [14, 13, 12, 12, 12, 12, 12, 11, 11],
                [14, 14, 13, 13, 13, 13, 13, 12, 12],
                [15, 14, 13, 13, 13, 12, 12, 12, 11]]
        args_int16 = [np.asarray(arg, dtype=np.int16) for arg in args]
        args_int32 = [np.asarray(arg, dtype=np.int32) for arg in args]
        args_uint8 = [np.asarray(arg, dtype=np.uint8) for arg in args]
        args_float64 = [np.asarray(arg, dtype=np.float64) for arg in args]

        res_int16 = stats.alexandergovern(*args_int16)
        res_int32 = stats.alexandergovern(*args_int32)
        res_uint8 = stats.alexandergovern(*args_uint8)
        res_float64 = stats.alexandergovern(*args_float64)

        assert (res_int16.pvalue == res_int32.pvalue ==
                res_uint8.pvalue == res_float64.pvalue)
        assert (res_int16.statistic == res_int32.statistic ==
                res_uint8.statistic == res_float64.statistic)

    @skip_xp_backends('jax.numpy', reason="Requires `_axis_nan_policy` decorator")
    @skip_xp_backends('dask.array', reason="Requires `_axis_nan_policy` decorator")
    @pytest.mark.parametrize('case',[([1, 2], []), ([1, 2], 2), ([1, 2], [2])])
    def test_too_small_inputs(self, case, xp):
        # input array is of size zero or too small
        dtype = xp_default_dtype(xp)
        case = xp.asarray(case[0], dtype=dtype), xp.asarray(case[1], dtype=dtype)
        with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
            res = stats.alexandergovern(*case)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning")
    def test_bad_inputs(self, xp):
        # inputs are not finite (infinity)
        samples = xp.asarray([1., 2.]), xp.asarray([xp.inf, xp.inf])
        res = stats.alexandergovern(*samples)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

    def test_compare_r(self, xp):
        '''
        Data generated in R with
        > set.seed(1)
        > library("onewaytests")
        > library("tibble")
        > y <- c(rnorm(40, sd=10),
        +        rnorm(30, sd=15),
        +        rnorm(20, sd=20))
        > x <- c(rep("one", times=40),
        +        rep("two", times=30),
        +        rep("eight", times=20))
        > x <- factor(x)
        > ag.test(y ~ x, tibble(y,x))

        Alexander-Govern Test (alpha = 0.05)
        -------------------------------------------------------------
        data : y and x

        statistic  : 1.359941
        parameter  : 2
        p.value    : 0.5066321

        Result     : Difference is not statistically significant.
        -------------------------------------------------------------
        Example adapted from:
        https://eval-serv2.metpsy.uni-jena.de/wiki-metheval-hp/index.php/R_FUN_Alexander-Govern

        '''
        one = [-6.264538107423324, 1.8364332422208225, -8.356286124100471,
               15.952808021377916, 3.295077718153605, -8.204683841180152,
               4.874290524284853, 7.383247051292173, 5.757813516534923,
               -3.0538838715635603, 15.11781168450848, 3.898432364114311,
               -6.2124058054180376, -22.146998871774997, 11.249309181431082,
               -0.4493360901523085, -0.16190263098946087, 9.438362106852992,
               8.212211950980885, 5.939013212175088, 9.189773716082183,
               7.821363007310671, 0.745649833651906, -19.89351695863373,
               6.198257478947102, -0.5612873952900078, -1.557955067053293,
               -14.707523838992744, -4.781500551086204, 4.179415601997024,
               13.58679551529044, -1.0278772734299553, 3.876716115593691,
               -0.5380504058290512, -13.770595568286065, -4.149945632996798,
               -3.942899537103493, -0.5931339671118566, 11.000253719838831,
               7.631757484575442]

        two = [-2.4678539438038034, -3.8004252020476135, 10.454450631071062,
               8.34994798010486, -10.331335418242798, -10.612427354431794,
               5.468729432052455, 11.527993867731237, -1.6851931822534207,
               13.216615896813222, 5.971588205506021, -9.180395898761569,
               5.116795371366372, -16.94044644121189, 21.495355525515556,
               29.7059984775879, -5.508322146997636, -15.662019394747961,
               8.545794411636193, -2.0258190582123654, 36.024266407571645,
               -0.5886000409975387, 10.346090436761651, 0.4200323817099909,
               -11.14909813323608, 2.8318844927151434, -27.074379433365568,
               21.98332292344329, 2.2988000731784655, 32.58917505543229]

        eight = [9.510190577993251, -14.198928618436291, 12.214527069781099,
                 -18.68195263288503, -25.07266800478204, 5.828924710349257,
                 -8.86583746436866, 0.02210703263248262, 1.4868264830332811,
                 -11.79041892376144, -11.37337465637004, -2.7035723024766414,
                 23.56173993146409, -30.47133600859524, 11.878923752568431,
                 6.659007424270365, 21.261996745527256, -6.083678472686013,
                 7.400376198325763, 5.341975815444621]

        one, two, eight = xp.asarray(one), xp.asarray(two), xp.asarray(eight)
        soln = stats.alexandergovern(one, two, eight)
        xp_assert_close(soln.statistic, xp.asarray(1.3599405447999450836))
        xp_assert_close(soln.pvalue, xp.asarray(0.50663205309676440091))

    def test_compare_scholar(self, xp):
        '''
        Data taken from 'The Modification and Evaluation of the
        Alexander-Govern Test in Terms of Power' by Kingsley Ochuko, T.,
        Abdullah, S., Binti Zain, Z., & Soaad Syed Yahaya, S. (2015).
        '''
        young = [482.43, 484.36, 488.84, 495.15, 495.24, 502.69, 504.62,
                 518.29, 519.1, 524.1, 524.12, 531.18, 548.42, 572.1, 584.68,
                 609.09, 609.53, 666.63, 676.4]
        middle = [335.59, 338.43, 353.54, 404.27, 437.5, 469.01, 485.85,
                  487.3, 493.08, 494.31, 499.1, 886.41]
        old = [519.01, 528.5, 530.23, 536.03, 538.56, 538.83, 557.24, 558.61,
               558.95, 565.43, 586.39, 594.69, 629.22, 645.69, 691.84]
        young, middle, old = xp.asarray(young), xp.asarray(middle), xp.asarray(old)
        soln = stats.alexandergovern(young, middle, old)
        xp_assert_close(soln.statistic, xp.asarray(5.3237), atol=1e-3)
        xp_assert_close(soln.pvalue, xp.asarray(0.06982), atol=1e-4)

        # verify with ag.test in r
        '''
        > library("onewaytests")
        > library("tibble")
        > young <- c(482.43, 484.36, 488.84, 495.15, 495.24, 502.69, 504.62,
        +                  518.29, 519.1, 524.1, 524.12, 531.18, 548.42, 572.1,
        +                  584.68, 609.09, 609.53, 666.63, 676.4)
        > middle <- c(335.59, 338.43, 353.54, 404.27, 437.5, 469.01, 485.85,
        +                   487.3, 493.08, 494.31, 499.1, 886.41)
        > old <- c(519.01, 528.5, 530.23, 536.03, 538.56, 538.83, 557.24,
        +                   558.61, 558.95, 565.43, 586.39, 594.69, 629.22,
        +                   645.69, 691.84)
        > young_fct <- c(rep("young", times=19))
        > middle_fct <-c(rep("middle", times=12))
        > old_fct <- c(rep("old", times=15))
        > ag.test(a ~ b, tibble(a=c(young, middle, old), b=factor(c(young_fct,
        +                                              middle_fct, old_fct))))

        Alexander-Govern Test (alpha = 0.05)
        -------------------------------------------------------------
        data : a and b

        statistic  : 5.324629
        parameter  : 2
        p.value    : 0.06978651

        Result     : Difference is not statistically significant.
        -------------------------------------------------------------

        '''
        xp_assert_close(soln.statistic, xp.asarray(5.324629), rtol=2e-6)
        xp_assert_close(soln.pvalue, xp.asarray(0.06978651), rtol=2e-6)

    def test_compare_scholar3(self, xp):
        '''
        Data taken from 'Robustness And Comparative Power Of WelchAspin,
        Alexander-Govern And Yuen Tests Under Non-Normality And Variance
        Heteroscedasticity', by Ayed A. Almoied. 2017. Page 34-37.
        https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=2775&context=oa_dissertations
        '''
        x1 = [-1.77559, -1.4113, -0.69457, -0.54148, -0.18808, -0.07152,
              0.04696, 0.051183, 0.148695, 0.168052, 0.422561, 0.458555,
              0.616123, 0.709968, 0.839956, 0.857226, 0.929159, 0.981442,
              0.999554, 1.642958]
        x2 = [-1.47973, -1.2722, -0.91914, -0.80916, -0.75977, -0.72253,
              -0.3601, -0.33273, -0.28859, -0.09637, -0.08969, -0.01824,
              0.260131, 0.289278, 0.518254, 0.683003, 0.877618, 1.172475,
              1.33964, 1.576766]
        x1, x2 = xp.asarray(x1), xp.asarray(x2)
        soln = stats.alexandergovern(x1, x2)
        xp_assert_close(soln.statistic, xp.asarray(0.713526), atol=1e-5)
        xp_assert_close(soln.pvalue, xp.asarray(0.398276), atol=1e-5)

        '''
        tested in ag.test in R:
        > library("onewaytests")
        > library("tibble")
        > x1 <- c(-1.77559, -1.4113, -0.69457, -0.54148, -0.18808, -0.07152,
        +          0.04696, 0.051183, 0.148695, 0.168052, 0.422561, 0.458555,
        +          0.616123, 0.709968, 0.839956, 0.857226, 0.929159, 0.981442,
        +          0.999554, 1.642958)
        > x2 <- c(-1.47973, -1.2722, -0.91914, -0.80916, -0.75977, -0.72253,
        +         -0.3601, -0.33273, -0.28859, -0.09637, -0.08969, -0.01824,
        +         0.260131, 0.289278, 0.518254, 0.683003, 0.877618, 1.172475,
        +         1.33964, 1.576766)
        > x1_fact <- c(rep("x1", times=20))
        > x2_fact <- c(rep("x2", times=20))
        > a <- c(x1, x2)
        > b <- factor(c(x1_fact, x2_fact))
        > ag.test(a ~ b, tibble(a, b))
        Alexander-Govern Test (alpha = 0.05)
        -------------------------------------------------------------
        data : a and b

        statistic  : 0.7135182
        parameter  : 1
        p.value    : 0.3982783

        Result     : Difference is not statistically significant.
        -------------------------------------------------------------
        '''
        xp_assert_close(soln.statistic, xp.asarray(0.7135182), rtol=2e-6)
        xp_assert_close(soln.pvalue, xp.asarray(0.3982783), rtol=2e-6)

    def test_nan_policy_propagate(self, xp):
        args = xp.asarray([1., 2., 3., 4.]), xp.asarray([1, xp.nan])
        # default nan_policy is 'propagate'
        res = stats.alexandergovern(*args)
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))

    @skip_xp_backends('jax.numpy', reason="Requires `_axis_nan_policy` decorator")
    @skip_xp_backends('dask.array', reason="Requires `_axis_nan_policy` decorator")
    def test_nan_policy_raise(self, xp):
        args = xp.asarray([1., 2., 3., 4.]), xp.asarray([1., xp.nan])
        with assert_raises(ValueError, match="The input contains nan values"):
            stats.alexandergovern(*args, nan_policy='raise')

    @skip_xp_backends('jax.numpy', reason="Requires `_axis_nan_policy` decorator")
    @skip_xp_backends('dask.array', reason="Requires `_axis_nan_policy` decorator")
    def test_nan_policy_omit(self, xp):
        args_nan = xp.asarray([1, 2, 3, xp.nan, 4]), xp.asarray([1, xp.nan, 19, 25])
        args_no_nan = xp.asarray([1, 2, 3, 4]), xp.asarray([1, 19, 25])
        res_nan = stats.alexandergovern(*args_nan, nan_policy='omit')
        res_no_nan = stats.alexandergovern(*args_no_nan)
        xp_assert_equal(res_nan.pvalue, res_no_nan.pvalue)
        xp_assert_equal(res_nan.statistic, res_no_nan.statistic)

    @skip_xp_backends('jax.numpy', reason="Requires `_axis_nan_policy` decorator")
    @skip_xp_backends('dask.array', reason="Requires `_axis_nan_policy` decorator")
    def test_constant_input(self, xp):
        # Zero variance input, consistent with `stats.pearsonr`
        x1 = xp.asarray([0.667, 0.667, 0.667])
        x2 = xp.asarray([0.123, 0.456, 0.789])
        with pytest.warns(RuntimeWarning, match="Precision loss occurred..."):
            res = stats.alexandergovern(x1, x2)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

    @skip_xp_backends('jax.numpy', reason="Requires `_axis_nan_policy` decorator")
    @skip_xp_backends('dask.array', reason="Requires `_axis_nan_policy` decorator")
    @pytest.mark.parametrize('axis', [0, 1, None])
    def test_2d_input(self, xp, axis):
        # much more extensive testing for NumPy in `test_axis_nan_policy.py`
        rng = np.random.default_rng(23498389754392854235)
        shape = (11, 12)
        samples = [rng.random(shape) for i in range(3)]
        xp_samples = [xp.asarray(sample) for sample in samples]
        ref = stats.alexandergovern(*samples, axis=axis)
        res = stats.alexandergovern(*xp_samples, axis=axis)
        xp_assert_close(res.statistic, xp.asarray(ref.statistic))
        xp_assert_close(res.pvalue, xp.asarray(ref.pvalue))


@make_xp_test_case(stats.f_oneway)
class TestFOneWay:

    def test_trivial(self, xp):
        # A trivial test of stats.f_oneway, with F=0.
        F, p = stats.f_oneway(xp.asarray([0, 2]), xp.asarray([0, 2]))
        xp_assert_equal(F, xp.asarray(0.0))
        xp_assert_equal(p, xp.asarray(1.0))

    @pytest.mark.parametrize("dtype", [None, "float32", "float64"])
    def test_basic(self, dtype, xp):
        # Despite being a floating point calculation, this data should
        # result in F being exactly 2.0.
        if is_numpy(xp) and xp.__version__ < "2.0" and dtype=='float32':
            pytest.skip("NumPy doesn't preserve dtype pre-NEP 50.")
        dtype = None if dtype is None else getattr(xp, dtype)
        F, p = stats.f_oneway(xp.asarray([0, 2], dtype=dtype),
                              xp.asarray([2, 4], dtype=dtype))
        xp_assert_equal(F, xp.asarray(2.0, dtype=dtype))
        xp_assert_close(p, xp.asarray(1 - 0.5**0.5, dtype=dtype))

    @pytest.mark.parametrize("dtype", [None, "float32", "float64"])
    def test_unequal_var(self, dtype, xp):
        # toy samples with unequal variances and different observations
        if is_numpy(xp) and xp.__version__ < "2.0" and dtype=='float32':
            pytest.skip("NumPy doesn't preserve dtype pre-NEP 50.")
        dtype = None if dtype is None else getattr(xp, dtype)
        samples = [xp.asarray([-50.42, 40.31, -18.09, 35.58, -6.8, 0.22], dtype=dtype),
                   xp.asarray([23.44, 4.5, 15.1, 9.66], dtype=dtype),
                   xp.asarray([11.94, 11.1 , 9.87, 9.09, 3.33], dtype=dtype)]

        F, p = stats.f_oneway(*samples, equal_var=False)

        # R language as benchmark
        # group1 <- c(-50.42, 40.31, -18.09, 35.58, -6.8, 0.22)
        # group2 <- c(23.44, 4.5, 15.1, 9.66)
        # group3 <- c(11.94, 11.1 , 9.87, 9.09, 3.33)
        #
        # data <- data.frame(
        #     value = c(group1, group2, group3),
        #     group = factor(c(rep("G1", length(group1)),
        #                      rep("G2", length(group2)),
        #                      rep("G3", length(group3))))
        # )
        # welch_anova <- oneway.test(value ~ group, data = data, var.equal = FALSE)
        # welch_anova$statistic
        ## F: 0.609740409019517
        # welch_anova$p.value
        ## 0.574838941286302

        xp_assert_close(F, xp.asarray(0.609740409019517, dtype=dtype))
        xp_assert_close(p, xp.asarray(0.574838941286302, dtype=dtype))

    def test_equal_var_input_validation(self, xp):
        samples = [xp.asarray([-50.42, 40.31, -18.09, 35.58, -6.8, 0.22]),
                   xp.asarray([23.44, 4.5, 15.1, 9.66]),
                   xp.asarray([11.94, 11.1 , 9.87, 9.09, 3.33])]

        message = "Expected a boolean value for 'equal_var'"
        with pytest.raises(TypeError, match=message):
            stats.f_oneway(*samples, equal_var="False")

    def test_known_exact(self, xp):
        # Another trivial dataset for which the exact F and p can be
        # calculated on most platforms
        F, p = stats.f_oneway(xp.asarray([2]), xp.asarray([2]),
                              xp.asarray([2, 3, 4]))
        xp_assert_close(F, xp.asarray(3/5))  # assert_equal fails on some CI platforms
        xp_assert_close(p, xp.asarray(5/8))

    def test_large_integer_array(self, xp):
        a = xp.asarray([655, 788], dtype=xp.uint16)
        b = xp.asarray([789, 772], dtype=xp.uint16)
        F, p = stats.f_oneway(a, b)
        # The expected value was verified by computing it with mpmath with
        # 40 digits of precision.
        xp_assert_close(F, xp.asarray(0.77450216931805540))

    def test_result_attributes(self, xp):
        a = xp.asarray([655, 788], dtype=xp.uint16)
        b = xp.asarray([789, 772], dtype=xp.uint16)
        res = stats.f_oneway(a, b)
        attributes = ('statistic', 'pvalue')
        check_named_results(res, attributes, xp=xp)

    @pytest.mark.parametrize('test_case',
        ['SiRstv', 'SmLs01', 'SmLs02', 'SmLs03', 'AtmWtAg', 'SmLs04', 'SmLs05',
         'SmLs06', 'SmLs07', 'SmLs08', 'SmLs09'])
    def test_nist(self, test_case, xp):
        # These are the nist ANOVA files. They can be found at:
        # https://www.itl.nist.gov/div898/strd/anova/anova.html
        filename = test_case + ".dat"
        rtol = 1e-7
        fname = os.path.abspath(os.path.join(os.path.dirname(__file__),
                                             'data/nist_anova', filename))
        with open(fname) as f:
            content = f.read().split('\n')
        certified = [line.split() for line in content[40:48]
                     if line.strip()]
        dataf = np.loadtxt(fname, skiprows=60)
        y, x = dataf.T
        y = y.astype(int)
        caty = np.unique(y)
        f = float(certified[0][-1])

        xlist = [xp.asarray(x[y == i]) for i in caty]
        res = stats.f_oneway(*xlist)

        # With the hard test cases we relax the tolerance a bit.
        hard_tc = ('SmLs07', 'SmLs08', 'SmLs09')
        if test_case in hard_tc:
            rtol = 1e-4

        xp_assert_close(res[0], xp.asarray(f, dtype=xp.float64), rtol=rtol)

    @pytest.mark.filterwarnings('ignore')  # Dask emits NumPy warnings
    @pytest.mark.parametrize("a, b, expected", [
        (np.array([42, 42, 42]), np.array([7, 7, 7]), (np.inf, 0.)),
        (np.array([42, 42, 42]), np.array([42, 42, 42]), (np.nan, np.nan))
        ])
    def test_constant_input(self, a, b, expected, xp):
        # For more details, look on https://github.com/scipy/scipy/issues/11669
        f, p = stats.f_oneway(xp.asarray(a), xp.asarray(b))
        xp_assert_equal(f, xp.asarray(expected[0]))
        xp_assert_equal(p, xp.asarray(expected[1]))

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy -> no _axis_nan_policy')
    @pytest.mark.skip_xp_backends('dask.array', reason='lazy -> no _axis_nan_policy')
    @pytest.mark.parametrize('axis', [-2, -1, 0, 1])
    def test_2d_inputs(self, axis, xp):
        a = np.array([[1, 4, 3, 3],
                      [2, 5, 3, 3],
                      [3, 6, 3, 3],
                      [2, 3, 3, 3],
                      [1, 4, 3, 3]], dtype=np.float64)
        b = np.array([[3, 1, 5, 3],
                      [4, 6, 5, 3],
                      [4, 3, 5, 3],
                      [1, 5, 5, 3],
                      [5, 5, 5, 3],
                      [2, 3, 5, 3],
                      [8, 2, 5, 3],
                      [2, 2, 5, 3]], dtype=np.float64)
        c = np.array([[4, 3, 4, 3],
                      [4, 2, 4, 3],
                      [5, 4, 4, 3],
                      [5, 4, 4, 3]], dtype=np.float64)

        if axis in [-1, 1]:
            a = a.T
            b = b.T
            c = c.T
            take_axis = 0
        else:
            take_axis = 1

        f, p = stats.f_oneway(xp.asarray(a), xp.asarray(b), xp.asarray(c),
                              axis=axis)

        # Verify that the result computed with the 2d arrays matches
        # the result of calling f_oneway individually on each slice.
        for j in [0, 1]:
            fj, pj = stats.f_oneway(np.take(a, j, take_axis),
                                    np.take(b, j, take_axis),
                                    np.take(c, j, take_axis))
            xp_assert_close(f[j], xp.asarray(fj))
            xp_assert_close(p[j], xp.asarray(pj))
        for j in [2, 3]:
            fj, pj = stats.f_oneway(np.take(a, j, take_axis),
                                    np.take(b, j, take_axis),
                                    np.take(c, j, take_axis))
            xp_assert_close(f[j], xp.asarray(fj))
            xp_assert_close(p[j], xp.asarray(pj))

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy -> no _axis_nan_policy')
    @pytest.mark.skip_xp_backends('dask.array', reason='lazy -> no _axis_nan_policy')
    def test_3d_inputs(self, xp):
        # Some 3-d arrays. (There is nothing special about the values.)
        a = xp.reshape(1/xp.arange(1.0, 4*5*7 + 1., dtype=xp.float64), (4, 5, 7))
        b = xp.reshape(2/xp.arange(1.0, 4*8*7 + 1., dtype=xp.float64), (4, 8, 7))
        c = xp.reshape(1/xp.arange(1.0, 4*4*7 + 1., dtype=xp.float64), (4, 4, 7))
        c = xp.cos(c)

        f, p = stats.f_oneway(a, b, c, axis=1)

        assert f.shape == (4, 7)
        assert p.shape == (4, 7)

        for i in range(a.shape[0]):
            for j in range(a.shape[2]):
                fij, pij = stats.f_oneway(a[i, :, j], b[i, :, j], c[i, :, j])
                xp_assert_close(fij, f[i, j])
                xp_assert_close(pij, p[i, j])

    def test_length0_1d_error(self, xp):
        # Require at least one value in each group.
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            result = stats.f_oneway(xp.asarray([1., 2., 3.]), xp.asarray([]),
                                    xp.asarray([4., 5., 6., 7.]))
            xp_assert_equal(result.statistic, xp.asarray(xp.nan))
            xp_assert_equal(result.pvalue, xp.asarray(xp.nan))

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy -> no _axis_nan_policy')
    @pytest.mark.skip_xp_backends('dask.array', reason='lazy -> no _axis_nan_policy')
    def test_length0_2d_error(self, xp):
        with eager_warns(SmallSampleWarning, match=too_small_nd_not_omit, xp=xp):
            ncols = 3
            a = xp.ones((4, ncols))
            b = xp.ones((0, ncols))
            c = xp.ones((5, ncols))
            f, p = stats.f_oneway(a, b, c)
            nans = xp.full((ncols,), fill_value=xp.nan)
            xp_assert_equal(f, nans)
            xp_assert_equal(p, nans)

    def test_all_length_one(self, xp):
        samples = xp.asarray([10]), xp.asarray([11]), xp.asarray([12]), xp.asarray([13])
        with eager_warns(SmallSampleWarning, xp=xp):
            result = stats.f_oneway(*samples)
            xp_assert_equal(result.statistic, xp.asarray(xp.nan))
            xp_assert_equal(result.pvalue, xp.asarray(xp.nan))

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy->reduced input validation')
    @pytest.mark.skip_xp_backends('dask.array', reason='lazy->reduced input validation')
    @pytest.mark.parametrize('args', [(), ([1, 2, 3],)])
    def test_too_few_inputs(self, args, xp):
        args = [xp.asarray(arg) for arg in args]
        message = "At least two samples are required..."
        with pytest.raises(TypeError, match=message):
            stats.f_oneway(*args)

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy->reduced input validation')
    @pytest.mark.skip_xp_backends('dask.array', reason='lazy->reduced input validation')
    def test_axis_error(self, xp):
        a = xp.ones((3, 4))
        b = xp.ones((5, 4))
        with pytest.raises(AxisError):
            stats.f_oneway(a, b, axis=2)

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy->reduced input validation')
    @pytest.mark.skip_xp_backends('dask.array', reason='lazy->reduced input validation')
    def test_bad_shapes(self, xp):
        a = xp.ones((3, 4))
        b = xp.ones((5, 4))
        with pytest.raises(ValueError):
            stats.f_oneway(a, b, axis=1)


@make_xp_test_case(stats.kruskal)
class TestKruskal:
    def test_array_like(self):
        h, p = stats.kruskal([1], [2])
        assert_equal(h, 1.0)
        assert_allclose(p, stats.chi2.sf(h, 1))

    def test_simple(self, xp):
        x = xp.asarray([1])
        y = xp.asarray([2])
        h, p = stats.kruskal(x, y)
        xp_assert_close(h, xp.asarray(1.0))
        dtype = xp_default_dtype(xp)
        xp_assert_close(p, xp.asarray(stats.chi2.sf(1, 1), dtype=dtype))

    @pytest.mark.parametrize("dtype", ['float32', 'float64', None])
    def test_basic(self, xp, dtype):
        if dtype == 'float32' and np.__version__ < "2":
            pytest.skip("Scalar dtypes only respected after NEP 50.")
        dtype = xp_default_dtype(xp) if dtype is None else getattr(xp, dtype)
        x = xp.asarray([1, 3, 5, 7, 9], dtype=dtype)
        y = xp.asarray([2, 4, 6, 8, 10], dtype=dtype)
        h, p = stats.kruskal(x, y)
        xp_assert_close(h, xp.asarray(3/11, dtype=dtype))
        xp_assert_close(p, xp.asarray(stats.chi2.sf(3/11, 1), dtype=dtype))

    def test_simple_tie(self, xp):
        x = [1]
        y = [1, 2]
        h, p = stats.kruskal(xp.asarray(x), xp.asarray(y))

        h_uncorr = 1.5**2 + 2*2.25**2 - 12
        corr = 0.75
        expected = xp.asarray(h_uncorr / corr)   # 0.5

        # Since the expression is simple and the exact answer is 0.5, it
        # should be safe to use assert_equal().
        xp_assert_equal(h, expected)
        xp_assert_close(p, special.chdtrc(xp.asarray(1.), expected))

    def test_another_tie(self, xp):
        x = [1, 1, 1, 2]
        y = [2, 2, 2, 2]
        h, p = stats.kruskal(xp.asarray(x), xp.asarray(y))

        h_uncorr = (12. / 8. / 9.) * 4 * (3**2 + 6**2) - 3 * 9
        corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
        expected = xp.asarray(h_uncorr / corr)

        xp_assert_close(h, expected)
        xp_assert_close(p, special.chdtrc(xp.asarray(1.), expected))

    def test_three_groups(self, xp):
        # A test of stats.kruskal with three groups, with ties.
        x = [1, 1, 1]
        y = [2, 2, 2]
        z = [2, 2]
        h, p = stats.kruskal(xp.asarray(x), xp.asarray(y), xp.asarray(z))

        h_uncorr = (12. / 8. / 9.) * (3*2**2 + 3*6**2 + 2*6**2) - 3 * 9  # 5.0
        corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
        expected = xp.asarray(h_uncorr / corr)  # 7.0

        xp_assert_close(h, expected)
        xp_assert_close(p, special.chdtrc(xp.asarray(2.), expected))

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy -> no _axis_nan_policy')
    def test_empty(self, xp):
        # A test of stats.kruskal with three groups, with ties.
        x = xp.asarray([1, 1, 1])
        y = xp.asarray([2, 2, 2])
        z = xp.asarray([], dtype=y.dtype)
        with pytest.warns(SmallSampleWarning, match=too_small_1d_not_omit):
            res = stats.kruskal(x, y, z)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

    @pytest.mark.skip_xp_backends('jax.numpy', reason='lazy -> no _axis_nan_policy')
    def test_nan_policy(self, xp):
        x = xp.arange(10.)
        x[9] = xp.nan

        res = stats.kruskal(x, x)
        xp_assert_equal(res.statistic, xp.asarray(xp.nan))
        xp_assert_equal(res.pvalue, xp.asarray(xp.nan))

        res = stats.kruskal(x, x, nan_policy='omit')
        xp_assert_equal(res.statistic, xp.asarray(0.0))
        xp_assert_equal(res.pvalue, xp.asarray(1.0))

        with pytest.raises(ValueError, match='The input contains nan values'):
            stats.kruskal(x, x, nan_policy='raise')

        with pytest.raises(ValueError, match='nan_policy must be one of...'):
            stats.kruskal(x, x, nan_policy='foobar')

    def test_large_samples(self, xp):
        # Test to see if large samples are handled correctly.
        n = 50000
        rng = np.random.default_rng(1808365978)
        x = xp.asarray(rng.standard_normal(n))
        y = xp.asarray(rng.standard_normal(n) + 50)
        h, p = stats.kruskal(x, y)
        xp_assert_close(p, xp.asarray(0., dtype=x.dtype))

    def test_no_args_gh20661(self, xp):
        message = r"Need at least two groups in stats.kruskal\(\)"
        with pytest.raises(ValueError, match=message):
            stats.kruskal()


@make_xp_test_case(stats.combine_pvalues)
class TestCombinePvalues:
    # Reference values computed using the following R code:
    # options(digits=16)
    # library(metap)
    # x = c(0.01, 0.2, 0.3)
    # sumlog(x)  # fisher
    # sumz(x)  # stouffer
    # sumlog(1-x)  # pearson (negative statistic and complement of p-value)
    # minimump(x)  # tippett
    @pytest.mark.parametrize(
        "method, expected_statistic, expected_pvalue",
        [("fisher", 14.83716180549625 , 0.02156175132483465),
         ("stouffer", 2.131790594240385, 0.01651203260896294),
         ("pearson", -1.179737662212887, 1-0.9778736999143087),
         ("tippett", 0.01, 0.02970100000000002),
         # mudholkar_george: library(transite); p_combine(x, method="MG")
         ("mudholkar_george", 6.828712071641684, 0.01654551838539527)])
    def test_reference_values(self, xp, method, expected_statistic, expected_pvalue):
        x = [.01, .2, .3]
        res = stats.combine_pvalues(xp.asarray(x), method=method)
        xp_assert_close(res.statistic, xp.asarray(expected_statistic))
        xp_assert_close(res.pvalue, xp.asarray(expected_pvalue))

    @pytest.mark.parametrize(
        # Reference values computed using R `metap` `sumz`:
        # options(digits=16)
        # library(metap)
        # x = c(0.01, 0.2, 0.3)
        # sumz(x, weights=c(1., 1., 1.))
        # sumz(x, weights=c(1., 4., 9.))
        "weights, expected_statistic, expected_pvalue",
        [([1., 1., 1.], 2.131790594240385, 0.01651203260896294),
         ([1., 4., 9.], 1.051815015753598, 0.1464422142261314)])
    def test_weighted_stouffer(self, xp, weights, expected_statistic, expected_pvalue):
        x = xp.asarray([.01, .2, .3])
        res = stats.combine_pvalues(x, method='stouffer', weights=xp.asarray(weights))
        xp_assert_close(res.statistic, xp.asarray(expected_statistic))
        xp_assert_close(res.pvalue, xp.asarray(expected_pvalue))

    methods = ["fisher", "pearson", "tippett", "stouffer", "mudholkar_george"]

    @pytest.mark.parametrize("variant", ["single", "all", "random"])
    @pytest.mark.parametrize("method", methods)
    def test_monotonicity(self, variant, method, xp):
        # Test that result increases monotonically with respect to input.
        m, n = 10, 7
        rng = np.random.default_rng(278448169958891062669391462690811630763)

        # `pvaluess` is an m × n array of p values. Each row corresponds to
        # a set of p values to be combined with p values increasing
        # monotonically down one column (single), simultaneously down each
        # column (all), or independently down each column (random).
        if variant == "single":
            pvaluess = xp.broadcast_to(xp.asarray(rng.random(n)), (m, n))
            pvaluess = xp.concat([xp.reshape(xp.linspace(0.1, 0.9, m), (-1, 1)),
                                  pvaluess[:, 1:]], axis=1)
        elif variant == "all":
            pvaluess = xp.broadcast_to(xp.linspace(0.1, 0.9, m), (n, m)).T
        elif variant == "random":
            pvaluess = xp.sort(xp.asarray(rng.uniform(0, 1, size=(m, n))), axis=0)

        combined_pvalues = xp.asarray([
            float(stats.combine_pvalues(pvaluess[i, :], method=method)[1])
            for i in range(pvaluess.shape[0])
        ])
        assert xp.all(combined_pvalues[1:] - combined_pvalues[:-1] >= 0)

    @pytest.mark.parametrize("method", methods)
    def test_result(self, method, xp):
        res = stats.combine_pvalues(xp.asarray([.01, .2, .3]), method=method)
        xp_assert_equal(res.statistic, res[0])
        xp_assert_equal(res.pvalue, res[1])

    @pytest.mark.parametrize("method", methods)
    @pytest.mark.parametrize("axis", [0, 1, None])
    def test_axis(self, method, axis, xp):
        rng = np.random.default_rng(234892349810482)
        x = xp.asarray(rng.random(size=(2, 10)))
        x = x.T if (axis == 0) else x
        res = stats.combine_pvalues(x, axis=axis, method=method)

        if axis is None:
            x = xp.reshape(x, (-1,))
            ref = stats.combine_pvalues(x, method=method)
            xp_assert_close(res.statistic, ref.statistic)
            xp_assert_close(res.pvalue, ref.pvalue)
            return

        x = x.T if (axis == 0) else x
        x0, x1 = x[0, :], x[1, :]
        ref0 = stats.combine_pvalues(x0, method=method)
        ref1 = stats.combine_pvalues(x1, method=method)

        xp_assert_close(res.statistic[0], ref0.statistic)
        xp_assert_close(res.statistic[1], ref1.statistic)
        xp_assert_close(res.pvalue[0], ref0.pvalue)
        xp_assert_close(res.pvalue[1], ref1.pvalue)


class TestCdfDistanceValidation:
    """
    Test that _cdf_distance() (via wasserstein_distance()) raises ValueErrors
    for bad inputs.
    """

    def test_distinct_value_and_weight_lengths(self):
        # When the number of weights does not match the number of values,
        # a ValueError should be raised.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [1], [2], [4], [3, 1])
        assert_raises(ValueError, stats.wasserstein_distance, [1], [2], [1, 0])

    def test_zero_weight(self):
        # When a distribution is given zero weight, a ValueError should be
        # raised.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [0, 1], [2], [0, 0])
        assert_raises(ValueError, stats.wasserstein_distance,
                      [0, 1], [2], [3, 1], [0])

    def test_negative_weights(self):
        # A ValueError should be raised if there are any negative weights.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [0, 1], [2, 2], [1, 1], [3, -1])

    def test_empty_distribution(self):
        # A ValueError should be raised when trying to measure the distance
        # between something and nothing.
        assert_raises(ValueError, stats.wasserstein_distance, [], [2, 2])
        assert_raises(ValueError, stats.wasserstein_distance, [1], [])

    def test_inf_weight(self):
        # An inf weight is not valid.
        assert_raises(ValueError, stats.wasserstein_distance,
                      [1, 2, 1], [1, 1], [1, np.inf, 1], [1, 1])


class TestWassersteinDistanceND:
    """ Tests for wasserstein_distance_nd() output values.
    """

    def test_published_values(self):
        # Compare against published values and manually computed results.
        # The values and computed result are posted at James D. McCaffrey's blog,
        # https://jamesmccaffrey.wordpress.com/2018/03/05/earth-mover-distance
        # -wasserstein-metric-example-calculation/
        u = [(1,1), (1,1), (1,1), (1,1), (1,1), (1,1), (1,1), (1,1), (1,1), (1,1),
             (4,2), (6,1), (6,1)]
        v = [(2,1), (2,1), (3,2), (3,2), (3,2), (5,1), (5,1), (5,1), (5,1), (5,1),
             (5,1), (5,1), (7,1)]

        res = stats.wasserstein_distance_nd(u, v)
        # In original post, the author kept two decimal places for ease of calculation.
        # This test uses the more precise value of distance to get the precise results.
        # For comparison, please see the table and figure in the original blog post.
        flow = np.array([2., 3., 5., 1., 1., 1.])
        dist = np.array([1.00, 5**0.5, 4.00, 2**0.5, 1.00, 1.00])
        ref = np.sum(flow * dist)/np.sum(flow)
        assert_allclose(res, ref)

    @pytest.mark.parametrize('n_value', (4, 15, 35))
    @pytest.mark.parametrize('ndim', (3, 4, 7))
    @pytest.mark.parametrize('max_repeats', (5, 10))
    def test_same_distribution_nD(self, ndim, n_value, max_repeats):
        # Any distribution moved to itself should have a Wasserstein distance
        # of zero.
        rng = np.random.default_rng(363836384995579937222333)
        repeats = rng.integers(1, max_repeats, size=n_value, dtype=int)

        u_values = rng.random(size=(n_value, ndim))
        v_values = np.repeat(u_values, repeats, axis=0)
        v_weights = rng.random(np.sum(repeats))
        range_repeat = np.repeat(np.arange(len(repeats)), repeats)
        u_weights = np.bincount(range_repeat, weights=v_weights)
        index = rng.permutation(len(v_weights))
        v_values, v_weights = v_values[index], v_weights[index]

        res = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)
        assert_allclose(res, 0, atol=1e-15)

    @pytest.mark.parametrize('nu', (8, 9, 38))
    @pytest.mark.parametrize('nv', (8, 12, 17))
    @pytest.mark.parametrize('ndim', (3, 5, 23))
    def test_collapse_nD(self, nu, nv, ndim):
        # test collapse for n dimensional values
        # Collapsing a n-D distribution to a point distribution at zero
        # is equivalent to taking the average of the norm of data.
        rng = np.random.default_rng(38573488467338826109)
        u_values = rng.random(size=(nu, ndim))
        v_values = np.zeros((nv, ndim))
        u_weights = rng.random(size=nu)
        v_weights = rng.random(size=nv)
        ref = np.average(np.linalg.norm(u_values, axis=1), weights=u_weights)
        res = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)
        assert_allclose(res, ref)

    @pytest.mark.parametrize('nu', (8, 16, 32))
    @pytest.mark.parametrize('nv', (8, 16, 32))
    @pytest.mark.parametrize('ndim', (1, 2, 6))
    def test_zero_weight_nD(self, nu, nv, ndim):
        # Values with zero weight have no impact on the Wasserstein distance.
        rng = np.random.default_rng(38573488467338826109)
        u_values = rng.random(size=(nu, ndim))
        v_values = rng.random(size=(nv, ndim))
        u_weights = rng.random(size=nu)
        v_weights = rng.random(size=nv)
        ref = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)

        add_row, nrows = rng.integers(0, nu, size=2)
        add_value = rng.random(size=(nrows, ndim))
        u_values = np.insert(u_values, add_row, add_value, axis=0)
        u_weights = np.insert(u_weights, add_row, np.zeros(nrows), axis=0)
        res = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)
        assert_allclose(res, ref)

    def test_inf_values(self):
        # Inf values can lead to an inf distance or trigger a RuntimeWarning
        # (and return NaN) if the distance is undefined.
        uv, vv, uw = [[1, 1], [2, 1]], [[np.inf, -np.inf]], [1, 1]
        distance = stats.wasserstein_distance_nd(uv, vv, uw)
        assert_equal(distance, np.inf)
        with np.errstate(invalid='ignore'):
            uv, vv = [[np.inf, np.inf]], [[np.inf, -np.inf]]
            distance = stats.wasserstein_distance_nd(uv, vv)
            assert_equal(distance, np.nan)

    @pytest.mark.parametrize('nu', (10, 15, 20))
    @pytest.mark.parametrize('nv', (10, 15, 20))
    @pytest.mark.parametrize('ndim', (1, 3, 5))
    def test_multi_dim_nD(self, nu, nv, ndim):
        # Adding dimension on distributions do not affect the result
        rng = np.random.default_rng(2736495738494849509)
        u_values = rng.random(size=(nu, ndim))
        v_values = rng.random(size=(nv, ndim))
        u_weights = rng.random(size=nu)
        v_weights = rng.random(size=nv)
        ref = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)

        add_dim = rng.integers(0, ndim)
        add_value = rng.random()

        u_values = np.insert(u_values, add_dim, add_value, axis=1)
        v_values = np.insert(v_values, add_dim, add_value, axis=1)
        res = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)
        assert_allclose(res, ref)

    @pytest.mark.parametrize('nu', (7, 13, 19))
    @pytest.mark.parametrize('nv', (7, 13, 19))
    @pytest.mark.parametrize('ndim', (2, 4, 7))
    def test_orthogonal_nD(self, nu, nv, ndim):
        # orthogonal transformations do not affect the result of the
        # wasserstein_distance
        rng = np.random.default_rng(34746837464536)
        u_values = rng.random(size=(nu, ndim))
        v_values = rng.random(size=(nv, ndim))
        u_weights = rng.random(size=nu)
        v_weights = rng.random(size=nv)
        ref = stats.wasserstein_distance_nd(u_values, v_values, u_weights, v_weights)

        dist = stats.ortho_group(ndim)
        transform = dist.rvs(random_state=rng)
        shift = rng.random(size=ndim)
        res = stats.wasserstein_distance_nd(u_values @ transform + shift,
                                         v_values @ transform + shift,
                                         u_weights, v_weights)
        assert_allclose(res, ref)

    def test_error_code(self):
        rng = np.random.default_rng(52473644737485644836320101)
        with pytest.raises(ValueError, match='Invalid input values. The inputs'):
            u_values = rng.random(size=(4, 10, 15))
            v_values = rng.random(size=(6, 2, 7))
            _ = stats.wasserstein_distance_nd(u_values, v_values)
        with pytest.raises(ValueError, match='Invalid input values. Dimensions'):
            u_values = rng.random(size=(15,))
            v_values = rng.random(size=(3, 15))
            _ = stats.wasserstein_distance_nd(u_values, v_values)
        with pytest.raises(ValueError,
            match='Invalid input values. If two-dimensional'):
            u_values = rng.random(size=(2, 10))
            v_values = rng.random(size=(2, 2))
            _ = stats.wasserstein_distance_nd(u_values, v_values)

    @pytest.mark.parametrize('u_size', [1, 10, 50])
    @pytest.mark.parametrize('v_size', [1, 10, 50])
    def test_optimization_vs_analytical(self, u_size, v_size):
        rng = np.random.default_rng(45634745675)
        # Test when u_weights = None, v_weights = None
        u_values = rng.random(size=(u_size, 1))
        v_values = rng.random(size=(v_size, 1))
        u_values_flat = u_values.ravel()
        v_values_flat = v_values.ravel()
        # These three calculations are done using different backends
        # but they must be equal
        d1 = stats.wasserstein_distance(u_values_flat, v_values_flat)
        d2 = stats.wasserstein_distance_nd(u_values, v_values)
        d3 = stats.wasserstein_distance_nd(u_values_flat, v_values_flat)
        assert_allclose(d2, d1)
        assert_allclose(d3, d1)
        # Test with u_weights and v_weights specified.
        u_weights = rng.random(size=u_size)
        v_weights = rng.random(size=v_size)
        d1 = stats.wasserstein_distance(u_values_flat, v_values_flat,
                                        u_weights, v_weights)
        d2 = stats.wasserstein_distance_nd(u_values, v_values,
                                        u_weights, v_weights)
        d3 = stats.wasserstein_distance_nd(u_values_flat, v_values_flat,
                                        u_weights, v_weights)
        assert_allclose(d2, d1)
        assert_allclose(d3, d1)


class TestWassersteinDistance:
    """ Tests for wasserstein_distance() output values.
    """

    def test_simple(self):
        # For basic distributions, the value of the Wasserstein distance is
        # straightforward.
        assert_allclose(
            stats.wasserstein_distance([0, 1], [0], [1, 1], [1]),
            .5)
        assert_allclose(stats.wasserstein_distance(
            [0, 1], [0], [3, 1], [1]),
            .25)
        assert_allclose(stats.wasserstein_distance(
            [0, 2], [0], [1, 1], [1]),
            1)
        assert_allclose(stats.wasserstein_distance(
            [0, 1, 2], [1, 2, 3]),
            1)

    def test_same_distribution(self):
        # Any distribution moved to itself should have a Wasserstein distance
        # of zero.
        assert_equal(stats.wasserstein_distance([1, 2, 3], [2, 1, 3]), 0)
        assert_equal(
            stats.wasserstein_distance([1, 1, 1, 4], [4, 1],
                                       [1, 1, 1, 1], [1, 3]),
            0)

    def test_shift(self):
        # If the whole distribution is shifted by x, then the Wasserstein
        # distance should be the norm of x.
        assert_allclose(stats.wasserstein_distance([0], [1]), 1)
        assert_allclose(stats.wasserstein_distance([-5], [5]), 10)
        assert_allclose(
            stats.wasserstein_distance([1, 2, 3, 4, 5], [11, 12, 13, 14, 15]),
            10)
        assert_allclose(
            stats.wasserstein_distance([4.5, 6.7, 2.1], [4.6, 7, 9.2],
                                       [3, 1, 1], [1, 3, 1]),
            2.5)

    def test_combine_weights(self):
        # Assigning a weight w to a value is equivalent to including that value
        # w times in the value array with weight of 1.
        assert_allclose(
            stats.wasserstein_distance(
                [0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
                [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
            stats.wasserstein_distance([5, 0, 1], [0, 4, 3],
                                       [1, 2, 4], [1, 2, 4]))

    def test_collapse(self):
        # Collapsing a distribution to a point distribution at zero is
        # equivalent to taking the average of the absolute values of the
        # values.
        u = np.arange(-10, 30, 0.3)
        v = np.zeros_like(u)
        assert_allclose(
            stats.wasserstein_distance(u, v),
            np.mean(np.abs(u)))

        u_weights = np.arange(len(u))
        v_weights = u_weights[::-1]
        assert_allclose(
            stats.wasserstein_distance(u, v, u_weights, v_weights),
            np.average(np.abs(u), weights=u_weights))

    def test_zero_weight(self):
        # Values with zero weight have no impact on the Wasserstein distance.
        assert_allclose(
            stats.wasserstein_distance([1, 2, 100000], [1, 1],
                                       [1, 1, 0], [1, 1]),
            stats.wasserstein_distance([1, 2], [1, 1], [1, 1], [1, 1]))

    def test_inf_values(self):
        # Inf values can lead to an inf distance or trigger a RuntimeWarning
        # (and return NaN) if the distance is undefined.
        assert_equal(
            stats.wasserstein_distance([1, 2, np.inf], [1, 1]),
            np.inf)
        assert_equal(
            stats.wasserstein_distance([1, 2, np.inf], [-np.inf, 1]),
            np.inf)
        assert_equal(
            stats.wasserstein_distance([1, -np.inf, np.inf], [1, 1]),
            np.inf)
        with pytest.warns(RuntimeWarning, match="invalid value"):
            assert_equal(
                stats.wasserstein_distance([1, 2, np.inf], [np.inf, 1]),
                np.nan)


class TestEnergyDistance:
    """ Tests for energy_distance() output values.
    """

    def test_simple(self):
        # For basic distributions, the value of the energy distance is
        # straightforward.
        assert_almost_equal(
            stats.energy_distance([0, 1], [0], [1, 1], [1]),
            np.sqrt(2) * .5)
        assert_almost_equal(stats.energy_distance(
            [0, 1], [0], [3, 1], [1]),
            np.sqrt(2) * .25)
        assert_almost_equal(stats.energy_distance(
            [0, 2], [0], [1, 1], [1]),
            2 * .5)
        assert_almost_equal(
            stats.energy_distance([0, 1, 2], [1, 2, 3]),
            np.sqrt(2) * (3*(1./3**2))**.5)

    def test_same_distribution(self):
        # Any distribution moved to itself should have a energy distance of
        # zero.
        assert_equal(stats.energy_distance([1, 2, 3], [2, 1, 3]), 0)
        assert_equal(
            stats.energy_distance([1, 1, 1, 4], [4, 1], [1, 1, 1, 1], [1, 3]),
            0)

    def test_shift(self):
        # If a single-point distribution is shifted by x, then the energy
        # distance should be sqrt(2) * sqrt(x).
        assert_almost_equal(stats.energy_distance([0], [1]), np.sqrt(2))
        assert_almost_equal(
            stats.energy_distance([-5], [5]),
            np.sqrt(2) * 10**.5)

    def test_combine_weights(self):
        # Assigning a weight w to a value is equivalent to including that value
        # w times in the value array with weight of 1.
        assert_almost_equal(
            stats.energy_distance([0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
                                  [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
            stats.energy_distance([5, 0, 1], [0, 4, 3], [1, 2, 4], [1, 2, 4]))

    def test_zero_weight(self):
        # Values with zero weight have no impact on the energy distance.
        assert_almost_equal(
            stats.energy_distance([1, 2, 100000], [1, 1], [1, 1, 0], [1, 1]),
            stats.energy_distance([1, 2], [1, 1], [1, 1], [1, 1]))

    def test_inf_values(self):
        # Inf values can lead to an inf distance or trigger a RuntimeWarning
        # (and return NaN) if the distance is undefined.
        assert_equal(stats.energy_distance([1, 2, np.inf], [1, 1]), np.inf)
        assert_equal(
            stats.energy_distance([1, 2, np.inf], [-np.inf, 1]),
            np.inf)
        assert_equal(
            stats.energy_distance([1, -np.inf, np.inf], [1, 1]),
            np.inf)
        with pytest.warns(RuntimeWarning, match="invalid value"):
            assert_equal(
                stats.energy_distance([1, 2, np.inf], [np.inf, 1]),
                np.nan)


@make_xp_test_case(stats.brunnermunzel)
class TestBrunnerMunzel:
    # Data from (Lumley, 1996)
    X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1]
    Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]

    def test_brunnermunzel_one_sided(self, xp):
        # Results are compared with R's lawstat package.
        X, Y = xp.asarray(self.X), xp.asarray(self.Y)
        u1, p1 = stats.brunnermunzel(X, Y, alternative='less')
        u2, p2 = stats.brunnermunzel(Y, X, alternative='greater')
        u3, p3 = stats.brunnermunzel(X, Y, alternative='greater')
        u4, p4 = stats.brunnermunzel(Y, X, alternative='less')

        xp_assert_close(p1, p2)
        xp_assert_close(p3, p4)
        assert p1 != p3
        xp_assert_close(u1, xp.asarray(3.1374674823029505))
        xp_assert_close(u2, xp.asarray(-3.1374674823029505))
        xp_assert_close(u3, xp.asarray(3.1374674823029505))
        xp_assert_close(u4, xp.asarray(-3.1374674823029505))

        xp_assert_close(p1, xp.asarray(0.0028931043330757342))
        xp_assert_close(p3, xp.asarray(0.99710689566692423))

    def test_brunnermunzel_two_sided(self, xp):
        # Results are compared with R's lawstat package.
        X, Y = xp.asarray(self.X), xp.asarray(self.Y)
        u1, p1 = stats.brunnermunzel(X, Y, alternative='two-sided')
        u2, p2 = stats.brunnermunzel(Y, X, alternative='two-sided')

        xp_assert_close(p1, xp.asarray(p2))
        xp_assert_close(u1, xp.asarray(3.1374674823029505))
        xp_assert_close(u2, xp.asarray(-3.1374674823029505))
        xp_assert_close(p1, xp.asarray(0.0057862086661515377))

    def test_brunnermunzel_default(self, xp):
        # The default value for alternative is two-sided
        X, Y = xp.asarray(self.X), xp.asarray(self.Y)
        u1, p1 = stats.brunnermunzel(X, Y)
        u2, p2 = stats.brunnermunzel(Y, X)

        xp_assert_close(p1, p2)
        xp_assert_close(u1, xp.asarray(3.1374674823029505))
        xp_assert_close(u2, xp.asarray(-3.1374674823029505))
        xp_assert_close(p1, xp.asarray(0.0057862086661515377))

    def test_brunnermunzel_alternative_error(self, xp):
        alternative = "error"
        distribution = "t"
        nan_policy = "propagate"
        assert alternative not in ["two-sided", "greater", "less"]
        message = "`alternative` must be 'less', 'greater', or 'two-sided'."
        with pytest.raises(ValueError, match=message):
            stats.brunnermunzel(xp.asarray(self.X), xp.asarray(self.Y),
                                alternative, distribution, nan_policy)

    def test_brunnermunzel_distribution_norm(self, xp):
        X, Y = xp.asarray(self.X), xp.asarray(self.Y)
        u1, p1 = stats.brunnermunzel(X, Y, distribution="normal")
        u2, p2 = stats.brunnermunzel(Y, X, distribution="normal")
        xp_assert_close(p1, xp.asarray(p2))
        xp_assert_close(u1, xp.asarray(3.1374674823029505))
        xp_assert_close(u2, xp.asarray(-3.1374674823029505))
        xp_assert_close(p1, xp.asarray(0.0017041417600383024))

    def test_brunnermunzel_distribution_error(self, xp):
        alternative = "two-sided"
        distribution = "error"
        nan_policy = "propagate"
        assert distribution not in ["t", "normal"]
        message = "distribution should be 't' or 'normal'"
        with pytest.raises(ValueError, match=message):
            stats.brunnermunzel(xp.asarray(self.X), xp.asarray(self.Y),
                                alternative, distribution, nan_policy)

    @pytest.mark.parametrize("kwarg_update", [{'y': []}, {'x': []},
                                              {'x': [], 'y': []}])
    def test_brunnermunzel_empty_imput(self, kwarg_update, xp):
        kwargs = {'x': self.X, 'y': self.Y}
        kwargs.update(kwarg_update)
        kwargs = {key:xp.asarray(val, dtype=xp_default_dtype(xp))
                  for key, val in kwargs.items()}
        with eager_warns(SmallSampleWarning, match=too_small_1d_not_omit, xp=xp):
            statistic, pvalue = stats.brunnermunzel(**kwargs)
        xp_assert_equal(statistic, xp.asarray(xp.nan))
        xp_assert_equal(pvalue, xp.asarray(xp.nan))

    def test_brunnermunzel_nan_input_propagate(self, xp):
        X = xp.asarray([1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, xp.nan])
        Y = xp.asarray([3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4.])
        u1, p1 = stats.brunnermunzel(X, Y, nan_policy="propagate")
        u2, p2 = stats.brunnermunzel(Y, X, nan_policy="propagate")

        xp_assert_equal(u1, xp.asarray(xp.nan))
        xp_assert_equal(p1, xp.asarray(xp.nan))
        xp_assert_equal(u2, xp.asarray(xp.nan))
        xp_assert_equal(p2, xp.asarray(xp.nan))

    def test_brunnermunzel_nan_input_raise(self, xp):
        X = xp.asarray([1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, xp.nan])
        Y = xp.asarray([3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4.])
        alternative = "two-sided"
        distribution = "t"
        nan_policy = "raise"

        message = "The input contains nan values"
        with pytest.raises(ValueError, match=message):
            stats.brunnermunzel(X, Y, alternative, distribution, nan_policy)
        with pytest.raises(ValueError, match=message):
            stats.brunnermunzel(Y, X, alternative, distribution, nan_policy)

    def test_brunnermunzel_nan_input_omit(self, xp):
        X = xp.asarray([1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan])
        Y = xp.asarray([3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4.])
        u1, p1 = stats.brunnermunzel(X, Y, nan_policy="omit")
        u2, p2 = stats.brunnermunzel(Y, X, nan_policy="omit")

        xp_assert_close(p1, p2)
        xp_assert_close(u1, xp.asarray(3.1374674823029505))
        xp_assert_close(u2, xp.asarray(-3.1374674823029505))
        xp_assert_close(p1, xp.asarray(0.0057862086661515377))

    @pytest.mark.parametrize("vectorized_call", [False, True])
    def test_brunnermunzel_return_nan(self, vectorized_call, xp):
        """ tests that a warning is emitted when p is nan
        p-value with t-distributions can be nan (0/0) (see gh-15843)
        """
        x = xp.asarray([1., 2., 3.])
        y = xp.asarray([5., 6., 7., 8., 9.])

        if vectorized_call:
            x = xp.stack((x, x)).T
            y = xp.stack((y, y)).T

        msg = "p-value cannot be estimated|divide by zero|invalid value encountered"
        with eager_warns(RuntimeWarning, match=msg, xp=xp):
            stats.brunnermunzel(x, y, distribution="t")

    def test_brunnermunzel_normal_dist(self, xp):
        """ tests that a p is 0 for datasets that cause p->nan
        when t-distribution is used (see gh-15843)
        """
        x = xp.asarray([1., 2., 3.])
        y = xp.asarray([5., 6., 7., 8., 9.])

        with eager_warns(RuntimeWarning, match='divide by zero', xp=xp):
            _, p = stats.brunnermunzel(x, y, distribution="normal")
        xp_assert_equal(p, xp.asarray(0.))


class TestQuantileTest:
    r""" Test the non-parametric quantile test,
    including the computation of confidence intervals
    """

    def test_quantile_test_iv(self):
        x = [1, 2, 3]

        message = "`x` must be a one-dimensional array of numbers."
        with pytest.raises(ValueError, match=message):
            stats.quantile_test([x])

        message = "`q` must be a scalar."
        with pytest.raises(ValueError, match=message):
            stats.quantile_test(x, q=[1, 2])

        message = "`p` must be a float strictly between 0 and 1."
        with pytest.raises(ValueError, match=message):
            stats.quantile_test(x, p=[0.5, 0.75])
        with pytest.raises(ValueError, match=message):
            stats.quantile_test(x, p=2)
        with pytest.raises(ValueError, match=message):
            stats.quantile_test(x, p=-0.5)

        message = "`alternative` must be one of..."
        with pytest.raises(ValueError, match=message):
            stats.quantile_test(x, alternative='one-sided')

        message = "`confidence_level` must be a number between 0 and 1."
        with pytest.raises(ValueError, match=message):
            stats.quantile_test(x).confidence_interval(1)

    @pytest.mark.parametrize(
        'p, alpha, lb, ub, alternative',
        [[0.3, 0.95, 1.221402758160170, 1.476980793882643, 'two-sided'],
         [0.5, 0.9, 1.506817785112854, 1.803988415397857, 'two-sided'],
         [0.25, 0.95, -np.inf, 1.39096812846378, 'less'],
         [0.8, 0.9, 2.117000016612675, np.inf, 'greater']]
    )
    def test_R_ci_quantile(self, p, alpha, lb, ub, alternative):
        # Test against R library `confintr` function `ci_quantile`, e.g.
        # library(confintr)
        # options(digits=16)
        # x <- exp(seq(0, 1, by = 0.01))
        # ci_quantile(x, q = 0.3)$interval
        # ci_quantile(x, q = 0.5, probs = c(0.05, 0.95))$interval
        # ci_quantile(x, q = 0.25, probs = c(0, 0.95))$interval
        # ci_quantile(x, q = 0.8, probs = c(0.1, 1))$interval
        x = np.exp(np.arange(0, 1.01, 0.01))
        res = stats.quantile_test(x, p=p, alternative=alternative)
        assert_allclose(res.confidence_interval(alpha), [lb, ub], rtol=1e-15)

    @pytest.mark.parametrize(
        'q, p, alternative, ref',
        [[1.2, 0.3, 'two-sided', 0.01515567517648],
         [1.8, 0.5, 'two-sided', 0.1109183496606]]
    )
    def test_R_pvalue(self, q, p, alternative, ref):
        # Test against R library `snpar` function `quant.test`, e.g.
        # library(snpar)
        # options(digits=16)
        # x < - exp(seq(0, 1, by=0.01))
        # quant.test(x, q=1.2, p=0.3, exact=TRUE, alternative='t')
        x = np.exp(np.arange(0, 1.01, 0.01))
        res = stats.quantile_test(x, q=q, p=p, alternative=alternative)
        assert_allclose(res.pvalue, ref, rtol=1e-12)

    @pytest.mark.parametrize('case', ['continuous', 'discrete'])
    @pytest.mark.parametrize('alternative', ['less', 'greater'])
    @pytest.mark.parametrize('alpha', [0.9, 0.95])
    def test_pval_ci_match(self, case, alternative, alpha):
        # Verify that the following statement holds:

        # The 95% confidence interval corresponding with alternative='less'
        # has -inf as its lower bound, and the upper bound `xu` is the greatest
        # element from the sample `x` such that:
        # `stats.quantile_test(x, q=xu, p=p, alternative='less').pvalue``
        # will be greater than 5%.

        # And the corresponding statement for the alternative='greater' case.

        seed = int((7**len(case) + len(alternative))*alpha)
        rng = np.random.default_rng(seed)
        if case == 'continuous':
            p, q = rng.random(size=2)
            rvs = rng.random(size=100)
        else:
            rvs = rng.integers(1, 11, size=100)
            p = rng.random()
            q = rng.integers(1, 11)

        res = stats.quantile_test(rvs, q=q, p=p, alternative=alternative)
        ci = res.confidence_interval(confidence_level=alpha)

        # select elements inside the confidence interval based on alternative
        if alternative == 'less':
            i_inside = rvs <= ci.high
        else:
            i_inside = rvs >= ci.low

        for x in rvs[i_inside]:
            res = stats.quantile_test(rvs, q=x, p=p, alternative=alternative)
            assert res.pvalue > 1 - alpha

        for x in rvs[~i_inside]:
            res = stats.quantile_test(rvs, q=x, p=p, alternative=alternative)
            assert res.pvalue < 1 - alpha

    def test_match_conover_examples(self):
        # Test against the examples in [1] (Conover Practical Nonparametric
        # Statistics Third Edition) pg 139

        # Example 1
        # Data is [189, 233, 195, 160, 212, 176, 231, 185, 199, 213, 202, 193,
        # 174, 166, 248]
        # Two-sided test of whether the upper quartile (p=0.75) equals 193
        # (q=193). Conover shows that 7 of the observations are less than or
        # equal to 193, and "for the binomial random variable Y, P(Y<=7) =
        # 0.0173", so the two-sided p-value is twice that, 0.0346.
        x = [189, 233, 195, 160, 212, 176, 231, 185, 199, 213, 202, 193,
             174, 166, 248]
        pvalue_expected = 0.0346
        res = stats.quantile_test(x, q=193, p=0.75, alternative='two-sided')
        assert_allclose(res.pvalue, pvalue_expected, rtol=1e-5)

        # Example 2
        # Conover doesn't give explicit data, just that 8 out of 112
        # observations are 60 or less. The test is whether the median time is
        # equal to 60 against the alternative that the median is greater than
        # 60. The p-value is calculated as P(Y<=8), where Y is again a binomial
        # distributed random variable, now with p=0.5 and n=112. Conover uses a
        # normal approximation, but we can easily calculate the CDF of the
        # binomial distribution.
        x = [59]*8 + [61]*(112-8)
        pvalue_expected = stats.binom(p=0.5, n=112).pmf(k=8)
        res = stats.quantile_test(x, q=60, p=0.5, alternative='greater')
        assert_allclose(res.pvalue, pvalue_expected, atol=1e-10)


class TestPageTrendTest:
    def setup_method(self):
        self.rng = np.random.default_rng(1808365978)

    # expected statistic and p-values generated using R at
    # https://rdrr.io/cran/cultevo/, e.g.
    # library(cultevo)
    # data = rbind(c(72, 47, 73, 35, 47, 96, 30, 59, 41, 36, 56, 49, 81, 43,
    #                   70, 47, 28, 28, 62, 20, 61, 20, 80, 24, 50),
    #              c(68, 52, 60, 34, 44, 20, 65, 88, 21, 81, 48, 31, 31, 67,
    #                69, 94, 30, 24, 40, 87, 70, 43, 50, 96, 43),
    #              c(81, 13, 85, 35, 79, 12, 92, 86, 21, 64, 16, 64, 68, 17,
    #                16, 89, 71, 43, 43, 36, 54, 13, 66, 51, 55))
    # result = page.test(data, verbose=FALSE)
    # Most test cases generated to achieve common critical p-values so that
    # results could be checked (to limited precision) against tables in
    # scipy.stats.page_trend_test reference [1]

    rng = np.random.default_rng(3113562111)
    data_3_25 = rng.random((3, 25))
    rng = np.random.default_rng(3113562111)
    data_10_26 = rng.random((10, 26))

    ts = [
          (12949, 0.275539045444, False, 'asymptotic', data_3_25),
          (47221, 0.5703651063709, False, 'asymptotic', data_10_26),
          (12332, 0.7722477197436702, False, 'asymptotic',
           [[72, 47, 73, 35, 47, 96, 30, 59, 41, 36, 56, 49, 81,
             43, 70, 47, 28, 28, 62, 20, 61, 20, 80, 24, 50],
            [68, 52, 60, 34, 44, 20, 65, 88, 21, 81, 48, 31, 31,
             67, 69, 94, 30, 24, 40, 87, 70, 43, 50, 96, 43],
            [81, 13, 85, 35, 79, 12, 92, 86, 21, 64, 16, 64, 68,
             17, 16, 89, 71, 43, 43, 36, 54, 13, 66, 51, 55]]),
          (266, 4.121656378600823e-05, False, 'exact',
           [[1.5, 4., 8.3, 5, 19, 11],
            [5, 4, 3.5, 10, 20, 21],
            [8.4, 3.2, 10, 12, 14, 15]]),
          (332, 0.9566400920502488, True, 'exact',
           [[4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1],
            [4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1],
            [3, 4, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4],
            [1, 2, 3, 4], [1, 2, 3, 4]]),
          (241, 0.9622210164861476, True, 'exact',
           [[3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
            [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
            [3, 2, 1], [2, 1, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3],
            [1, 2, 3], [1, 2, 3], [1, 2, 3]]),
          (197, 0.9619432897162209, True, 'exact',
           [[6, 5, 4, 3, 2, 1], [6, 5, 4, 3, 2, 1], [1, 3, 4, 5, 2, 6]]),
          (423, 0.9590458306880073, True, 'exact',
           [[5, 4, 3, 2, 1], [5, 4, 3, 2, 1], [5, 4, 3, 2, 1],
            [5, 4, 3, 2, 1], [5, 4, 3, 2, 1], [5, 4, 3, 2, 1],
            [4, 1, 3, 2, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5],
            [1, 2, 3, 4, 5]]),
          (217, 0.9693058575034678, True, 'exact',
           [[3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
            [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
            [2, 1, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3],
            [1, 2, 3]]),
          (395, 0.991530289351305, True, 'exact',
           [[7, 6, 5, 4, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1],
            [6, 5, 7, 4, 3, 2, 1], [1, 2, 3, 4, 5, 6, 7]]),
          (117, 0.9997817843373017, True, 'exact',
           [[3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
            [3, 2, 1], [3, 2, 1], [3, 2, 1], [2, 1, 3], [1, 2, 3]]),
         ]

    @pytest.mark.parametrize("L, p, ranked, method, data", ts)
    def test_accuracy(self, L, p, ranked, method, data):
        res = stats.page_trend_test(data, ranked=ranked, method=method)
        assert_equal(L, res.statistic)
        assert_allclose(p, res.pvalue)
        assert_equal(method, res.method)

    ts2 = [
           (542, 0.9481266260876332, True, 'exact',
            [[10, 9, 8, 7, 6, 5, 4, 3, 2, 1],
             [1, 8, 4, 7, 6, 5, 9, 3, 2, 10]]),
           (1322, 0.9993113928199309, True, 'exact',
            [[10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 8, 7, 6, 5, 4, 3, 2, 1],
             [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [9, 2, 8, 7, 6, 5, 4, 3, 10, 1],
             [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]),
           (2286, 0.9908688345484833, True, 'exact',
            [[8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
             [8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
             [8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
             [8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
             [8, 7, 6, 5, 4, 3, 2, 1], [1, 3, 5, 6, 4, 7, 2, 8],
             [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8],
             [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8],
             [1, 2, 3, 4, 5, 6, 7, 8]]),
          ]
    # only the first of these appears slow because intermediate data are
    # cached and used on the rest
    @pytest.mark.parametrize("L, p, ranked, method, data", ts2)
    @pytest.mark.slow()
    def test_accuracy2(self, L, p, ranked, method, data):
        res = stats.page_trend_test(data, ranked=ranked, method=method)
        assert_equal(L, res.statistic)
        assert_allclose(p, res.pvalue)
        assert_equal(method, res.method)

    def test_options(self):
        rng = np.random.default_rng(183973867)
        m, n = 10, 20
        predicted_ranks = np.arange(1, n+1)
        perm = rng.permutation(np.arange(n))
        data = rng.random((m, n))
        ranks = stats.rankdata(data, axis=1)
        res1 = stats.page_trend_test(ranks)
        res2 = stats.page_trend_test(ranks, ranked=True)
        res3 = stats.page_trend_test(data, ranked=False)
        res4 = stats.page_trend_test(ranks, predicted_ranks=predicted_ranks)
        res5 = stats.page_trend_test(ranks[:, perm],
                                     predicted_ranks=predicted_ranks[perm])
        assert_equal(res1.statistic, res2.statistic)
        assert_equal(res1.statistic, res3.statistic)
        assert_equal(res1.statistic, res4.statistic)
        assert_equal(res1.statistic, res5.statistic)

    def test_Ames_assay(self):
        # test from _page_trend_test.py [2] page 151; data on page 144
        data = [[101, 117, 111], [91, 90, 107], [103, 133, 121],
                [136, 140, 144], [190, 161, 201], [146, 120, 116]]
        data = np.array(data).T
        predicted_ranks = np.arange(1, 7)

        res = stats.page_trend_test(data, ranked=False,
                                    predicted_ranks=predicted_ranks,
                                    method="asymptotic")
        assert_equal(res.statistic, 257)
        assert_almost_equal(res.pvalue, 0.0035, decimal=4)

        res = stats.page_trend_test(data, ranked=False,
                                    predicted_ranks=predicted_ranks,
                                    method="exact")
        assert_equal(res.statistic, 257)
        assert_almost_equal(res.pvalue, 0.0023, decimal=4)

    def test_input_validation(self):
        # test data not a 2d array
        with assert_raises(ValueError, match="`data` must be a 2d array."):
            stats.page_trend_test(None)
        with assert_raises(ValueError, match="`data` must be a 2d array."):
            stats.page_trend_test([])
        with assert_raises(ValueError, match="`data` must be a 2d array."):
            stats.page_trend_test([1, 2])
        with assert_raises(ValueError, match="`data` must be a 2d array."):
            stats.page_trend_test([[[1]]])

        # test invalid dimensions
        rng = np.random.default_rng(2482566048)
        with assert_raises(ValueError, match="Page's L is only appropriate"):
            stats.page_trend_test(rng.random((1, 3)))
        with assert_raises(ValueError, match="Page's L is only appropriate"):
            stats.page_trend_test(rng.random((2, 2)))

        # predicted ranks must include each integer [1, 2, 3] exactly once
        message = "`predicted_ranks` must include each integer"
        with assert_raises(ValueError, match=message):
            stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                  predicted_ranks=[0, 1, 2])
        with assert_raises(ValueError, match=message):
            stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                  predicted_ranks=[1.1, 2, 3])
        with assert_raises(ValueError, match=message):
            stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                  predicted_ranks=[1, 2, 3, 3])
        with assert_raises(ValueError, match=message):
            stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                  predicted_ranks="invalid")

        # test improperly ranked data
        with assert_raises(ValueError, match="`data` is not properly ranked"):
            stats.page_trend_test([[0, 2, 3], [1, 2, 3]], True)
        with assert_raises(ValueError, match="`data` is not properly ranked"):
            stats.page_trend_test([[1, 2, 3], [1, 2, 4]], True)

        # various
        with assert_raises(ValueError, match="`data` contains NaNs"):
            stats.page_trend_test([[1, 2, 3], [1, 2, np.nan]],
                                  ranked=False)
        with assert_raises(ValueError, match="`method` must be in"):
            stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                  method="ekki")
        with assert_raises(TypeError, match="`ranked` must be boolean."):
            stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                  ranked="ekki")


rng = np.random.default_rng(902340982)
x = rng.random(10)
y = rng.random(10)


@pytest.mark.parametrize("fun, args",
                         [(stats.wilcoxon, (x,)),
                          (stats.ks_1samp, (x, stats.norm.cdf)),  # type: ignore[attr-defined]  # noqa: E501
                          (stats.ks_2samp, (x, y)),
                          (stats.kstest, (x, y)),
                          ])
def test_rename_mode_method(fun, args):

    res = fun(*args, method='exact')
    res2 = fun(*args, mode='exact')
    assert_equal(res, res2)

    err = rf"{fun.__name__}() got multiple values for argument"
    with pytest.raises(TypeError, match=re.escape(err)):
        fun(*args, method='exact', mode='exact')


class TestExpectile:
    def test_same_as_mean(self):
        rng = np.random.default_rng(42)
        x = rng.random(size=20)
        assert_allclose(stats.expectile(x, alpha=0.5), np.mean(x))

    def test_minimum(self):
        rng = np.random.default_rng(42)
        x = rng.random(size=20)
        assert_allclose(stats.expectile(x, alpha=0), np.amin(x))

    def test_maximum(self):
        rng = np.random.default_rng(42)
        x = rng.random(size=20)
        assert_allclose(stats.expectile(x, alpha=1), np.amax(x))

    def test_weights(self):
        # expectile should minimize `fun` defined below; see
        # F. Sobotka and T. Kneib, "Geoadditive expectile regression",
        # Computational Statistics and Data Analysis 56 (2012) 755-767
        # :doi:`10.1016/j.csda.2010.11.015`
        rng = np.random.default_rng(1856392524598679138)

        def fun(u, a, alpha, weights):
            w = np.full_like(a, fill_value=alpha)
            w[a <= u] = 1 - alpha
            return np.sum(w * weights * (a - u)**2)

        def expectile2(a, alpha, weights):
            bracket = np.min(a), np.max(a)
            return optimize.minimize_scalar(fun, bracket=bracket,
                                            args=(a, alpha, weights)).x

        n = 10
        a = rng.random(n)
        alpha = rng.random()
        weights = rng.random(n)

        res = stats.expectile(a, alpha, weights=weights)
        ref = expectile2(a, alpha, weights)
        assert_allclose(res, ref)

    @pytest.mark.parametrize(
        "alpha", [0.2, 0.5 - 1e-12, 0.5, 0.5 + 1e-12, 0.8]
    )
    @pytest.mark.parametrize("n", [20, 2000])
    def test_expectile_properties(self, alpha, n):
        """
        See Section 6 of
        I. Steinwart, C. Pasin, R.C. Williamson & S. Zhang (2014).
        "Elicitation and Identification of Properties". COLT.
        http://proceedings.mlr.press/v35/steinwart14.html

        and

        Propositions 5, 6, 7 of
        F. Bellini, B. Klar, and A. Müller and E. Rosazza Gianin (2013).
        "Generalized Quantiles as Risk Measures"
        http://doi.org/10.2139/ssrn.2225751
        """
        rng = np.random.default_rng(42)
        x = rng.normal(size=n)

        # 0. definite / constancy
        # Let T(X) denote the expectile of rv X ~ F.
        # T(c) = c for constant c
        for c in [-5, 0, 0.5]:
            assert_allclose(
                stats.expectile(np.full(shape=n, fill_value=c), alpha=alpha),
                c
            )

        # 1. translation equivariance
        # T(X + c) = T(X) + c
        c = rng.exponential()
        assert_allclose(
            stats.expectile(x + c, alpha=alpha),
            stats.expectile(x, alpha=alpha) + c,
        )
        assert_allclose(
            stats.expectile(x - c, alpha=alpha),
            stats.expectile(x, alpha=alpha) - c,
        )

        # 2. positively homogeneity
        # T(cX) = c * T(X) for c > 0
        assert_allclose(
            stats.expectile(c * x, alpha=alpha),
            c * stats.expectile(x, alpha=alpha),
        )

        # 3. subadditivity
        # Note that subadditivity holds for alpha >= 0.5.
        # T(X + Y) <= T(X) + T(Y)
        # For alpha = 0.5, i.e. the mean, strict equality holds.
        # For alpha < 0.5, one can use property 6. to show
        # T(X + Y) >= T(X) + T(Y)
        y = rng.logistic(size=n, loc=10)  # different distribution than x
        if alpha == 0.5:
            def assert_op(a, b):
                assert_allclose(a, b)

        elif alpha > 0.5:
            def assert_op(a, b):
                assert a < b

        else:
            def assert_op(a, b):
                assert a > b

        assert_op(
            stats.expectile(np.r_[x + y], alpha=alpha),
            stats.expectile(x, alpha=alpha)
            + stats.expectile(y, alpha=alpha)
        )

        # 4. monotonicity
        # This holds for first order stochastic dominance X:
        # X >= Y whenever P(X <= x) < P(Y <= x)
        # T(X) <= T(Y) whenever X <= Y
        y = rng.normal(size=n, loc=5)
        assert (
            stats.expectile(x, alpha=alpha) <= stats.expectile(y, alpha=alpha)
        )

        # 5. convexity for alpha > 0.5, concavity for alpha < 0.5
        # convexity is
        # T((1 - c) X + c Y) <= (1 - c) T(X) + c T(Y) for 0 <= c <= 1
        y = rng.logistic(size=n, loc=10)
        for c in [0.1, 0.5, 0.8]:
            assert_op(
                stats.expectile((1-c)*x + c*y, alpha=alpha),
                (1-c) * stats.expectile(x, alpha=alpha) +
                c * stats.expectile(y, alpha=alpha)
            )

        # 6. negative argument
        # T_{alpha}(-X) = -T_{1-alpha}(X)
        assert_allclose(
            stats.expectile(-x, alpha=alpha),
            -stats.expectile(x, alpha=1-alpha),
        )

    @pytest.mark.parametrize("n", [20, 2000])
    def test_monotonicity_in_alpha(self, n):
        rng = np.random.default_rng(42)
        x = rng.pareto(a=2, size=n)
        e_list = []
        alpha_seq = np.logspace(-15, np.log10(0.5), 100)
        # sorted list of unique alpha values in interval (0, 1)
        for alpha in np.r_[0, alpha_seq, 1 - alpha_seq[:-1:-1], 1]:
            e_list.append(stats.expectile(x, alpha=alpha))
        assert np.all(np.diff(e_list) > 0)


@make_xp_test_case(stats.lmoment)
class TestLMoment:
    # data from https://github.com/scipy/scipy/issues/19460
    data = [0.87, 0.87, 1.29, 1.5, 1.7, 0.66, 1.5, 0.5, 1., 1.25, 2.3,
            1.03, 2.85, 0.68, 1.74, 1.94, 0.63, 2.04, 1.2, 0.64, 2.05, 0.97,
            2.81, 1.02, 2.76, 0.86, 1.36, 1.29, 1.68, 0.72, 1.67, 1.15, 3.26,
            0.93, 0.83, 0.91, 0.92, 2.32, 1.12, 3.21, 1.23, 1.22, 1.29, 2.08,
            0.64, 2.83, 2.68, 1.77, 0.69, 1.69, 0.7, 1.83, 2.25, 1.23, 1.17,
            0.94, 1.22, 0.76, 0.69, 0.48, 1.04, 2.49, 1.38, 1.57, 1.79, 1.59,
            1.3, 1.54, 1.07, 1.03, 0.76, 2.35, 2.05, 2.02, 2.36, 1.59, 0.97,
            1.63, 1.66, 0.94, 1.45, 1.26, 1.25, 0.68, 2.96, 0.8, 1.16, 0.82,
            0.64, 0.87, 1.33, 1.28, 1.26, 1.19, 1.24, 1.12, 1.45, 1.03, 1.37,
            1.4, 1.35, 1.28, 1.04, 1.31, 0.87, 0.96, 2.55, 1.72, 1.05, 1.15,
            1.73, 1.03, 1.53, 2.41, 1.36, 2.08, 0.92, 0.73, 1.56, 1.94, 0.78]

    not_integers = [1.5, [1, 2, 3.5], math.nan, math.inf]

    def test_dtype_iv(self, xp):
        message = '`sample` must be an array of real numbers.'
        with pytest.raises(ValueError, match=message):
            stats.lmoment(xp.asarray(self.data, dtype=xp.complex128))

    @skip_xp_invalid_arg
    def test_dtype_iv_non_numeric(self):
        message = '`sample` must be an array of real numbers.'
        with pytest.raises(ValueError, match=message):
            stats.lmoment(np.array(self.data, dtype=object))

    @pytest.mark.parametrize('order', not_integers + [0, -1, [], [[1, 2, 3]]])
    def test_order_iv(self, order, xp):
        message = '`order` must be a scalar or a non-empty...'
        with pytest.raises(ValueError, match=message):
            stats.lmoment(xp.asarray(self.data), order=order)

    @pytest.mark.parametrize('axis', not_integers)
    def test_axis_iv(self, axis, xp):
        message = '`axis` must be an integer'
        with pytest.raises(ValueError, match=message):
            stats.lmoment(xp.asarray(self.data), axis=axis)

    @pytest.mark.parametrize('sorted', not_integers)
    def test_sorted_iv(self, sorted, xp):
        message = '`sorted` must be True or False.'
        with pytest.raises(ValueError, match=message):
            stats.lmoment(xp.asarray(self.data), sorted=sorted)

    @pytest.mark.parametrize('standardize', not_integers)
    def test_standardize_iv(self, standardize, xp):
        message = '`standardize` must be True or False.'
        with pytest.raises(ValueError, match=message):
            stats.lmoment(xp.asarray(self.data), standardize=standardize)

    @pytest.mark.parametrize('order', [1, 4, [1, 2, 3, 4]])
    @pytest.mark.parametrize('standardize', [False, True])
    @pytest.mark.parametrize('presorted', [False, True])
    def test_lmoment(self, order, standardize, presorted, xp):
        # Reference values from R package `lmom`
        # options(digits=16)
        # library(lmom)
        # data= c(0.87, 0.87,..., 1.94, 0.78)
        # samlmu(data)
        ref = xp.asarray([1.4087603305785130, 0.3415936639118458,
                          0.2189964482831403, 0.1328186463415905])

        if not standardize:
            ref = xpx.at(ref)[2:].multiply(ref[1])

        data = sorted(self.data) if presorted else self.data
        data = xp.asarray(data)

        res = stats.lmoment(data, order, standardize=standardize, sorted=presorted)
        xp_assert_close(res, ref[xp.asarray(order)-1])

    def test_dtype(self, xp):
        dtype = xp.float32
        sample = xp.asarray(self.data)
        res = stats.lmoment(xp.astype(sample, dtype))
        ref = xp.astype(stats.lmoment(sample), dtype)
        xp_assert_close(res, ref, rtol=1e-4)

        dtype = xp.int64
        sample = xp.asarray([1, 2, 3, 4, 5])
        res = stats.lmoment(xp.astype(sample, dtype))
        ref = stats.lmoment(xp.astype(sample, xp_default_dtype(xp)))
        xp_assert_close(res, ref)

    @pytest.mark.parametrize("axis", [0, 1])
    def test_axis(self, axis, xp):
        # nd input is tested extensively in `test_axis_nan_policy`, but only for NumPy
        rng = np.random.default_rng(234923498149931248151)
        x = rng.random(size=(10, 11))
        res = stats.lmoment(xp.asarray(x), axis=axis)
        ref = xp.asarray(stats.lmoment(x, axis=axis))
        xp_assert_close(res, ref)


class TestXP_Mean:
    @pytest.mark.parametrize('axis', [None, 1, -1, (-2, 2)])
    @pytest.mark.parametrize('weights', [None, True])
    @pytest.mark.parametrize('keepdims', [False, True])
    def test_xp_mean_basic(self, xp, axis, weights, keepdims):
        rng = np.random.default_rng(90359458245906)
        x = rng.random((3, 4, 5))
        x_xp = xp.asarray(x)
        w = w_xp = None

        if weights:
            w = rng.random((1, 5))
            w_xp = xp.asarray(w)
            x, w = np.broadcast_arrays(x, w)

        res = _xp_mean(x_xp, weights=w_xp, axis=axis, keepdims=keepdims)
        ref = np.average(x, weights=w, axis=axis, keepdims=keepdims)

        xp_assert_close(res, xp.asarray(ref))

    def test_non_broadcastable(self, xp):
        # non-broadcastable x and weights
        x, w = xp.arange(10.), xp.zeros(5)
        message = "Array shapes are incompatible for broadcasting."
        with pytest.raises(ValueError, match=message):
            _xp_mean(x, weights=w)

    @pytest.mark.filterwarnings("ignore:divide by zero encountered:RuntimeWarning:dask")
    @pytest.mark.filterwarnings("ignore:invalid value encountered:RuntimeWarning:dask")
    def test_special_cases(self, xp):
        # weights sum to zero
        weights = xp.asarray([-1., 0., 1.])

        res = _xp_mean(xp.asarray([1., 1., 1.]), weights=weights)
        xp_assert_close(res, xp.asarray(xp.nan))

        res = _xp_mean(xp.asarray([2., 1., 1.]), weights=weights)
        xp_assert_close(res, xp.asarray(-np.inf))

        res = _xp_mean(xp.asarray([1., 1., 2.]), weights=weights)
        xp_assert_close(res, xp.asarray(np.inf))

    @pytest.mark.filterwarnings(
        "ignore:invalid value encountered:RuntimeWarning"
    ) # for dask
    def test_nan_policy(self, xp):
        x = xp.arange(10.)
        mask = (x == 3)
        x = xp.where(mask, xp.nan, x)

        # nan_policy='raise' raises an error
        if is_lazy_array(x):
            with pytest.raises(TypeError, match='not supported for lazy arrays'):
                _xp_mean(x, nan_policy='raise')
        else:
            with pytest.raises(ValueError, match='The input contains nan values'):
                _xp_mean(x, nan_policy='raise')

        # `nan_policy='propagate'` is the default, and the result is NaN
        res1 = _xp_mean(x)
        res2 = _xp_mean(x, nan_policy='propagate')
        ref = xp.asarray(xp.nan)
        xp_assert_equal(res1, ref)
        xp_assert_equal(res2, ref)

        # `nan_policy='omit'` omits NaNs in `x`
        res = _xp_mean(x, nan_policy='omit')
        ref = xp.mean(x[~mask])
        xp_assert_close(res, ref)

        # `nan_policy='omit'` omits NaNs in `weights`, too
        weights = xp.ones(10)
        weights = xp.where(mask, xp.nan, weights)
        res = _xp_mean(xp.arange(10.), weights=weights, nan_policy='omit')
        ref = xp.mean(x[~mask])
        xp_assert_close(res, ref)

    @skip_xp_backends(eager_only=True)
    def test_nan_policy_warns(self, xp):
        x = xp.arange(10.)
        x = xp.where(x == 3, xp.nan, x)

        # Check for warning if omitting NaNs causes empty slice
        message = 'After omitting NaNs...'
        with pytest.warns(RuntimeWarning, match=message):
            res = _xp_mean(x * np.nan, nan_policy='omit')
            ref = xp.asarray(xp.nan)
            xp_assert_equal(res, ref)

    def test_empty(self, xp):
        message = 'One or more sample arguments is too small...'
        with pytest.warns(SmallSampleWarning, match=message):
            res = _xp_mean(xp.asarray([]))
            ref = xp.asarray(xp.nan)
            xp_assert_equal(res, ref)

        message = "All axis-slices of one or more sample arguments..."
        with pytest.warns(SmallSampleWarning, match=message):
            res = _xp_mean(xp.asarray([[]]), axis=1)
            ref = xp.asarray([xp.nan])
            xp_assert_equal(res, ref)

        res = _xp_mean(xp.asarray([[]]), axis=0)
        ref = xp.asarray([])
        xp_assert_equal(res, ref)

    @pytest.mark.filterwarnings(
        "ignore:overflow encountered in reduce:RuntimeWarning"
    ) # for dask
    def test_dtype(self, xp):
        max = xp.finfo(xp.float32).max
        x_np = np.asarray([max, max], dtype=np.float32)
        x_xp = xp.asarray(x_np)

        # Overflow occurs for float32 input
        with np.errstate(over='ignore'):
            res = _xp_mean(x_xp)
            ref = np.mean(x_np)
            np.testing.assert_equal(ref, np.inf)
            xp_assert_close(res, xp.asarray(ref))

        # correct result is returned if `float64` is used
        res = _xp_mean(x_xp, dtype=xp.float64)
        ref = xp.asarray(np.mean(np.asarray(x_np, dtype=np.float64)))
        xp_assert_close(res, ref)

    def test_integer(self, xp):
        # integer inputs are converted to the appropriate float
        x = xp.arange(10)
        y = xp.arange(10.)
        xp_assert_equal(_xp_mean(x), _xp_mean(y))
        xp_assert_equal(_xp_mean(y, weights=x), _xp_mean(y, weights=y))

    def test_complex_gh22404(self, xp):
        rng = np.random.default_rng(90359458245906)
        x, y, wx, wy = rng.random((4, 20))
        res = _xp_mean(xp.asarray(x + y*1j), weights=xp.asarray(wx + wy*1j))
        ref = np.average(x + y*1j, weights=wx + wy*1j)
        xp_assert_close(res, xp.asarray(ref))


class TestXP_Var:
    @pytest.mark.parametrize('axis', [None, 1, -1, (-2, 2)])
    @pytest.mark.parametrize('keepdims', [False, True])
    @pytest.mark.parametrize('correction', [0, 1])
    @pytest.mark.parametrize('nan_policy', ['propagate', 'omit'])
    def test_xp_var_basic(self, xp, axis, keepdims, correction, nan_policy):
        rng = np.random.default_rng(90359458245906)
        x = rng.random((3, 4, 5))
        var_ref = np.var

        if nan_policy == 'omit':
            nan_mask = rng.random(size=x.shape) > 0.5
            x[nan_mask] = np.nan
            var_ref = np.nanvar

        x_xp = xp.asarray(x)

        res = _xp_var(x_xp, axis=axis, keepdims=keepdims, correction=correction,
                      nan_policy=nan_policy)

        with warnings.catch_warnings():
            warnings.filterwarnings(
                "ignore", "Degrees of freedom <= 0 for slice", RuntimeWarning)
            ref = var_ref(x, axis=axis, keepdims=keepdims, ddof=correction)

        xp_assert_close(res, xp.asarray(ref))

    def test_special_cases(self, xp):
        # correction too big
        res = _xp_var(xp.asarray([1., 2.]), correction=3)
        xp_assert_close(res, xp.asarray(xp.nan))

    def test_nan_policy(self, xp):
        x = xp.arange(10.)
        mask = (x == 3)
        x = xp.where(mask, xp.nan, x)

        # `nan_policy='propagate'` is the default, and the result is NaN
        res1 = _xp_var(x)
        res2 = _xp_var(x, nan_policy='propagate')
        ref = xp.asarray(xp.nan)
        xp_assert_equal(res1, ref)
        xp_assert_equal(res2, ref)

        # `nan_policy='omit'` omits NaNs in `x`
        res = _xp_var(x, nan_policy='omit')
        ref = xp.var(x[~mask])
        xp_assert_close(res, ref)

    @skip_xp_backends(eager_only=True)
    def test_nan_policy_warns(self, xp):
        x = xp.arange(10.)
        x = xp.where(x == 3, xp.nan, x)

        # Check for warning if omitting NaNs causes empty slice
        message = 'After omitting NaNs...'
        with pytest.warns(RuntimeWarning, match=message):
            res = _xp_var(x * np.nan, nan_policy='omit')
            ref = xp.asarray(xp.nan)
            xp_assert_equal(res, ref)

    @skip_xp_backends(eager_only=True)
    def test_nan_policy_raise(self, xp):
        # nan_policy='raise' raises an error when NaNs are present
        message = 'The input contains nan values'
        with pytest.raises(ValueError, match=message):
            _xp_var(xp.asarray([1, 2, xp.nan]), nan_policy='raise')

    def test_empty(self, xp):
        message = 'One or more sample arguments is too small...'
        with pytest.warns(SmallSampleWarning, match=message):
            res = _xp_var(xp.asarray([]))
            ref = xp.asarray(xp.nan)
            xp_assert_equal(res, ref)

        message = "All axis-slices of one or more sample arguments..."
        with pytest.warns(SmallSampleWarning, match=message):
            res = _xp_var(xp.asarray([[]]), axis=1)
            ref = xp.asarray([xp.nan])
            xp_assert_equal(res, ref)

        res = _xp_var(xp.asarray([[]]), axis=0)
        ref = xp.asarray([])
        xp_assert_equal(res, ref)

    @pytest.mark.filterwarnings(
        "ignore:overflow encountered in reduce:RuntimeWarning"
    ) # Overflow occurs for float32 input
    def test_dtype(self, xp):
        max = xp.finfo(xp.float32).max
        x_np = np.asarray([max, max/2], dtype=np.float32)
        x_xp = xp.asarray(x_np)

        res = _xp_var(x_xp)
        ref = np.var(x_np)
        np.testing.assert_equal(ref, np.inf)
        xp_assert_close(res, xp.asarray(ref))

        # correct result is returned if `float64` is used
        res = _xp_var(x_xp, dtype=xp.float64)
        ref = xp.asarray(np.var(np.asarray(x_np, dtype=np.float64)))
        xp_assert_close(res, ref)

    def test_integer(self, xp):
        # integer inputs are converted to the appropriate float
        x = xp.arange(10)
        y = xp.arange(10.)
        xp_assert_equal(_xp_var(x), _xp_var(y))

    def test_complex_gh22404(self, xp):
        rng = np.random.default_rng(90359458245906)
        x, y = rng.random((2, 20))
        res = _xp_var(xp.asarray(x + y*1j))
        ref = np.var(x + y*1j)
        xp_assert_close(res, xp.asarray(ref), check_dtype=False)


def test_chk_asarray(xp):
    rng = np.random.default_rng(2348923425434)
    x0 = rng.random(size=(2, 3, 4))
    x = xp.asarray(x0)

    axis = 1
    x_out, axis_out = _chk_asarray(x, axis=axis, xp=xp)
    xp_assert_equal(x_out, xp.asarray(x0))
    assert_equal(axis_out, axis)

    axis = None
    x_out, axis_out = _chk_asarray(x, axis=axis, xp=xp)
    xp_assert_equal(x_out, xp.asarray(x0.ravel()))
    assert_equal(axis_out, 0)

    axis = 2
    x_out, axis_out = _chk_asarray(x[0, 0, 0], axis=axis, xp=xp)
    xp_assert_equal(x_out, xp.asarray(np.atleast_1d(x0[0, 0, 0])))
    assert_equal(axis_out, axis)