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

import pytest
import itertools
import warnings

from scipy import stats
from scipy.stats import (betabinom, betanbinom, hypergeom, nhypergeom,
                         bernoulli, boltzmann, skellam, zipf, zipfian, binom,
                         nbinom, nchypergeom_fisher, nchypergeom_wallenius,
                         randint, poisson_binom)

import numpy as np
from numpy.testing import (
    assert_almost_equal, assert_equal, assert_allclose
)
from scipy.special import binom as special_binom
from scipy.optimize import root_scalar
from scipy.integrate import quad


# The expected values were computed with Wolfram Alpha, using
# the expression CDF[HypergeometricDistribution[N, n, M], k].
@pytest.mark.parametrize('k, M, n, N, expected, rtol',
                         [(3, 10, 4, 5,
                           0.9761904761904762, 1e-15),
                          (107, 10000, 3000, 215,
                           0.9999999997226765, 1e-15),
                          (10, 10000, 3000, 215,
                           2.681682217692179e-21, 5e-11)])
def test_hypergeom_cdf(k, M, n, N, expected, rtol):
    p = hypergeom.cdf(k, M, n, N)
    assert_allclose(p, expected, rtol=rtol)


# The expected values were computed with Wolfram Alpha, using
# the expression SurvivalFunction[HypergeometricDistribution[N, n, M], k].
@pytest.mark.parametrize('k, M, n, N, expected, rtol',
                         [(25, 10000, 3000, 215,
                           0.9999999999052958, 1e-15),
                          (125, 10000, 3000, 215,
                           1.4416781705752128e-18, 5e-11)])
def test_hypergeom_sf(k, M, n, N, expected, rtol):
    p = hypergeom.sf(k, M, n, N)
    assert_allclose(p, expected, rtol=rtol)


def test_hypergeom_logpmf():
    # symmetries test
    # f(k,N,K,n) = f(n-k,N,N-K,n) = f(K-k,N,K,N-n) = f(k,N,n,K)
    k = 5
    N = 50
    K = 10
    n = 5
    logpmf1 = hypergeom.logpmf(k, N, K, n)
    logpmf2 = hypergeom.logpmf(n - k, N, N - K, n)
    logpmf3 = hypergeom.logpmf(K - k, N, K, N - n)
    logpmf4 = hypergeom.logpmf(k, N, n, K)
    assert_almost_equal(logpmf1, logpmf2, decimal=12)
    assert_almost_equal(logpmf1, logpmf3, decimal=12)
    assert_almost_equal(logpmf1, logpmf4, decimal=12)

    # test related distribution
    # Bernoulli distribution if n = 1
    k = 1
    N = 10
    K = 7
    n = 1
    hypergeom_logpmf = hypergeom.logpmf(k, N, K, n)
    bernoulli_logpmf = bernoulli.logpmf(k, K/N)
    assert_almost_equal(hypergeom_logpmf, bernoulli_logpmf, decimal=12)


def test_nhypergeom_pmf():
    # test with hypergeom
    M, n, r = 45, 13, 8
    k = 6
    NHG = nhypergeom.pmf(k, M, n, r)
    HG = hypergeom.pmf(k, M, n, k+r-1) * (M - n - (r-1)) / (M - (k+r-1))
    assert_allclose(HG, NHG, rtol=1e-10)


def test_nhypergeom_pmfcdf():
    # test pmf and cdf with arbitrary values.
    M = 8
    n = 3
    r = 4
    support = np.arange(n+1)
    pmf = nhypergeom.pmf(support, M, n, r)
    cdf = nhypergeom.cdf(support, M, n, r)
    assert_allclose(pmf, [1/14, 3/14, 5/14, 5/14], rtol=1e-13)
    assert_allclose(cdf, [1/14, 4/14, 9/14, 1.0], rtol=1e-13)


def test_nhypergeom_r0():
    # test with `r = 0`.
    M = 10
    n = 3
    r = 0
    pmf = nhypergeom.pmf([[0, 1, 2, 0], [1, 2, 0, 3]], M, n, r)
    assert_allclose(pmf, [[1, 0, 0, 1], [0, 0, 1, 0]], rtol=1e-13)


def test_nhypergeom_rvs_shape():
    # Check that when given a size with more dimensions than the
    # dimensions of the broadcast parameters, rvs returns an array
    # with the correct shape.
    x = nhypergeom.rvs(22, [7, 8, 9], [[12], [13]], size=(5, 1, 2, 3))
    assert x.shape == (5, 1, 2, 3)


def test_nhypergeom_accuracy():
    # Check that nhypergeom.rvs post-gh-13431 gives the same values as
    # inverse transform sampling
    rng = np.random.RandomState(0)
    x = nhypergeom.rvs(22, 7, 11, size=100, random_state=rng)
    rng = np.random.RandomState(0)
    p = rng.uniform(size=100)
    y = nhypergeom.ppf(p, 22, 7, 11)
    assert_equal(x, y)


def test_boltzmann_upper_bound():
    k = np.arange(-3, 5)

    N = 1
    p = boltzmann.pmf(k, 0.123, N)
    expected = k == 0
    assert_equal(p, expected)

    lam = np.log(2)
    N = 3
    p = boltzmann.pmf(k, lam, N)
    expected = [0, 0, 0, 4/7, 2/7, 1/7, 0, 0]
    assert_allclose(p, expected, rtol=1e-13)

    c = boltzmann.cdf(k, lam, N)
    expected = [0, 0, 0, 4/7, 6/7, 1, 1, 1]
    assert_allclose(c, expected, rtol=1e-13)


def test_betabinom_a_and_b_unity():
    # test limiting case that betabinom(n, 1, 1) is a discrete uniform
    # distribution from 0 to n
    n = 20
    k = np.arange(n + 1)
    p = betabinom(n, 1, 1).pmf(k)
    expected = np.repeat(1 / (n + 1), n + 1)
    assert_almost_equal(p, expected)


@pytest.mark.parametrize('dtypes', itertools.product(*[(int, float)]*3))
def test_betabinom_stats_a_and_b_integers_gh18026(dtypes):
    # gh-18026 reported that `betabinom` kurtosis calculation fails when some
    # parameters are integers. Check that this is resolved.
    n_type, a_type, b_type = dtypes
    n, a, b = n_type(10), a_type(2), b_type(3)
    assert_allclose(betabinom.stats(n, a, b, moments='k'), -0.6904761904761907)


def test_betabinom_bernoulli():
    # test limiting case that betabinom(1, a, b) = bernoulli(a / (a + b))
    a = 2.3
    b = 0.63
    k = np.arange(2)
    p = betabinom(1, a, b).pmf(k)
    expected = bernoulli(a / (a + b)).pmf(k)
    assert_almost_equal(p, expected)


def test_issue_10317():
    alpha, n, p = 0.9, 10, 1
    assert_equal(nbinom.interval(confidence=alpha, n=n, p=p), (0, 0))


def test_issue_11134():
    alpha, n, p = 0.95, 10, 0
    assert_equal(binom.interval(confidence=alpha, n=n, p=p), (0, 0))


def test_issue_7406():
    rng = np.random.default_rng(4763112764)
    assert_equal(binom.ppf(rng.random(10), 0, 0.5), 0)

    # Also check that endpoints (q=0, q=1) are correct
    assert_equal(binom.ppf(0, 0, 0.5), -1)
    assert_equal(binom.ppf(1, 0, 0.5), 0)


def test_issue_5122():
    rng = np.random.default_rng(8312492117)
    p = 0
    n = rng.integers(100, size=10)

    x = 0
    ppf = binom.ppf(x, n, p)
    assert_equal(ppf, -1)

    x = np.linspace(0.01, 0.99, 10)
    ppf = binom.ppf(x, n, p)
    assert_equal(ppf, 0)

    x = 1
    ppf = binom.ppf(x, n, p)
    assert_equal(ppf, n)


def test_issue_1603():
    assert_equal(binom(1000, np.logspace(-3, -100)).ppf(0.01), 0)


def test_issue_5503():
    p = 0.5
    x = np.logspace(3, 14, 12)
    assert_allclose(binom.cdf(x, 2*x, p), 0.5, atol=1e-2)


@pytest.mark.parametrize('x, n, p, cdf_desired', [
    (300, 1000, 3/10, 0.51559351981411995636),
    (3000, 10000, 3/10, 0.50493298381929698016),
    (30000, 100000, 3/10, 0.50156000591726422864),
    (300000, 1000000, 3/10, 0.50049331906666960038),
    (3000000, 10000000, 3/10, 0.50015600124585261196),
    (30000000, 100000000, 3/10, 0.50004933192735230102),
    (30010000, 100000000, 3/10, 0.98545384016570790717),
    (29990000, 100000000, 3/10, 0.01455017177985268670),
    (29950000, 100000000, 3/10, 5.02250963487432024943e-28),
    (300_000_000, 1_000_000_000, 3/10, 0.50001560012523938),
    (3_000_000_000, 10_000_000_000, 3/10, 0.50000493319275)
])
def test_issue_5503pt2(x, n, p, cdf_desired):
    assert_allclose(binom.cdf(x, n, p), cdf_desired)


def test_issue_5503pt3():
    # From Wolfram Alpha: CDF[BinomialDistribution[1e12, 1e-12], 2]
    assert_allclose(binom.cdf(2, 10**12, 10**-12), 0.91969860292869777384)


def test_issue_6682():
    # Reference value from R:
    # options(digits=16)
    # print(pnbinom(250, 50, 32/63, lower.tail=FALSE))
    assert_allclose(nbinom.sf(250, 50, 32./63.), 1.460458510976452e-35)


def test_issue_19747():
    # test that negative k does not raise an error in nbinom.logcdf
    result = nbinom.logcdf([5, -1, 1], 5, 0.5)
    reference = [-0.47313352, -np.inf, -2.21297293]
    assert_allclose(result, reference)


def test_boost_divide_by_zero_issue_15101():
    n = 1000
    p = 0.01
    k = 996
    assert_allclose(binom.pmf(k, n, p), 0.0)


def test_skellam_gh11474():
    # test issue reported in gh-11474 caused by `cdfchn`
    mu = [1, 10, 100, 1000, 5000, 5050, 5100, 5250, 6000]
    cdf = skellam.cdf(0, mu, mu)
    # generated in R
    # library(skellam)
    # options(digits = 16)
    # mu = c(1, 10, 100, 1000, 5000, 5050, 5100, 5250, 6000)
    # pskellam(0, mu, mu, TRUE)
    cdf_expected = [0.6542541612768356, 0.5448901559424127, 0.5141135799745580,
                    0.5044605891382528, 0.5019947363350450, 0.5019848365953181,
                    0.5019750827993392, 0.5019466621805060, 0.5018209330219539]
    assert_allclose(cdf, cdf_expected)


class TestZipfian:
    def test_zipfian_asymptotic(self):
        # test limiting case that zipfian(a, n) -> zipf(a) as n-> oo
        a = 6.5
        N = 10000000
        k = np.arange(1, 21)
        assert_allclose(zipfian.pmf(k, a, N), zipf.pmf(k, a))
        assert_allclose(zipfian.cdf(k, a, N), zipf.cdf(k, a))
        assert_allclose(zipfian.sf(k, a, N), zipf.sf(k, a))
        assert_allclose(zipfian.stats(a, N, moments='msvk'),
                        zipf.stats(a, moments='msvk'))

    def test_zipfian_continuity(self):
        # test that zipfian(0.999999, n) ~ zipfian(1.000001, n)
        # (a = 1 switches between methods of calculating harmonic sum)
        alt1, agt1 = 0.99999999, 1.00000001
        N = 30
        k = np.arange(1, N + 1)
        assert_allclose(zipfian.pmf(k, alt1, N), zipfian.pmf(k, agt1, N),
                        rtol=5e-7)
        assert_allclose(zipfian.cdf(k, alt1, N), zipfian.cdf(k, agt1, N),
                        rtol=5e-7)
        assert_allclose(zipfian.sf(k, alt1, N), zipfian.sf(k, agt1, N),
                        rtol=5e-7)
        assert_allclose(zipfian.stats(alt1, N, moments='msvk'),
                        zipfian.stats(agt1, N, moments='msvk'), rtol=5e-7)

    def test_zipfian_R(self):
        # test against R VGAM package
        # library(VGAM)
        # k <- c(13, 16,  1,  4,  4,  8, 10, 19,  5,  7)
        # a <- c(1.56712977, 3.72656295, 5.77665117, 9.12168729, 5.79977172,
        #        4.92784796, 9.36078764, 4.3739616 , 7.48171872, 4.6824154)
        # n <- c(70, 80, 48, 65, 83, 89, 50, 30, 20, 20)
        # pmf <- dzipf(k, N = n, shape = a)
        # cdf <- pzipf(k, N = n, shape = a)
        # print(pmf)
        # print(cdf)
        rng = np.random.RandomState(0)
        k = rng.randint(1, 20, size=10)
        a = rng.rand(10)*10 + 1
        n = rng.randint(1, 100, size=10)
        pmf = [8.076972e-03, 2.950214e-05, 9.799333e-01, 3.216601e-06,
               3.158895e-04, 3.412497e-05, 4.350472e-10, 2.405773e-06,
               5.860662e-06, 1.053948e-04]
        cdf = [0.8964133, 0.9998666, 0.9799333, 0.9999995, 0.9998584,
               0.9999458, 1.0000000, 0.9999920, 0.9999977, 0.9998498]
        # skip the first point; zipUC is not accurate for low a, n
        assert_allclose(zipfian.pmf(k, a, n)[1:], pmf[1:], rtol=1e-6)
        assert_allclose(zipfian.cdf(k, a, n)[1:], cdf[1:], rtol=5e-5)

    rng = np.random.RandomState(0)
    naive_tests = np.vstack((np.logspace(-2, 1, 10),
                             rng.randint(2, 40, 10))).T

    @pytest.mark.parametrize("a, n", naive_tests)
    def test_zipfian_naive(self, a, n):
        # test against bare-bones implementation

        @np.vectorize
        def Hns(n, s):
            """Naive implementation of harmonic sum"""
            return (1/np.arange(1, n+1)**s).sum()

        @np.vectorize
        def pzip(k, a, n):
            """Naive implementation of zipfian pmf"""
            if k < 1 or k > n:
                return 0.
            else:
                return 1 / k**a / Hns(n, a)

        k = np.arange(n+1)
        pmf = pzip(k, a, n)
        cdf = np.cumsum(pmf)
        mean = np.average(k, weights=pmf)
        var = np.average((k - mean)**2, weights=pmf)
        std = var**0.5
        skew = np.average(((k-mean)/std)**3, weights=pmf)
        kurtosis = np.average(((k-mean)/std)**4, weights=pmf) - 3
        assert_allclose(zipfian.pmf(k, a, n), pmf)
        assert_allclose(zipfian.cdf(k, a, n), cdf)
        assert_allclose(zipfian.stats(a, n, moments="mvsk"),
                        [mean, var, skew, kurtosis])

    def test_pmf_integer_k(self):
        k = np.arange(0, 1000)
        k_int32 = k.astype(np.int32)
        dist = zipfian(111, 22)
        pmf = dist.pmf(k)
        pmf_k_int32 = dist.pmf(k_int32)
        assert_equal(pmf, pmf_k_int32)

    # Reference values were computed with mpmath.
    @pytest.mark.parametrize(
        'k, a, n, ref',
        [(3, 1 + 1e-12, 10, 0.11380571738244807),
         (995, 1 + 1e-9, 1000, 0.0001342634472310051)]
    )
    def test_pmf_against_mpmath(self, k, a, n, ref):
        p = zipfian.pmf(k, a, n)
        assert_allclose(p, ref, 5e-16)

    # Reference values were computed with mpmath.
    @pytest.mark.parametrize(
        'k, a, n, ref',
        [(4990, 1.25, 5000, 5.780138225335147e-05),
         (9998, 3.5, 10000, 1.775352757966365e-14),
         (50000, 3.5, 100000, 5.227803621367486e-13),
         (10, 6.0, 100, 1.523108153557902e-06),
         (95, 6.0, 100, 5.572438078308601e-12)]
    )
    def test_sf_against_mpmath(self, k, a, n, ref):
        sf = zipfian.sf(k, a, n)
        assert_allclose(sf, ref, rtol=8e-15)

    # Reference values were computed with mpmath.
    @pytest.mark.parametrize(
        'a, n, ref',
        [(1 + 1e-9, 100, 19.27756356649707),
         (1 + 1e-7, 100000, 8271.194454731552),
         (1.001, 3, 1.6360225521183804)]
    )
    def test_mean_against_mpmath(self, a, n, ref):
        m = zipfian.mean(a, n)
        assert_allclose(m, ref, rtol=8e-15)

    def test_ridiculously_large_n(self):
        # This should return nan with no errors or warnings.
        a = 2.5
        n = 1e100
        p = zipfian.pmf(10, a, n)
        assert_equal(p, np.nan)


class TestNCH:
    def setup_method(self):
        rng = np.random.default_rng(7162434334)
        shape = (2, 4, 3)
        max_m = 100
        m1 = rng.integers(1, max_m, size=shape)  # red balls
        m2 = rng.integers(1, max_m, size=shape)  # white balls
        N = m1 + m2  # total balls
        n = randint.rvs(0, N, size=N.shape, random_state=rng)  # number of draws
        xl = np.maximum(0, n-m2)  # lower bound of support
        xu = np.minimum(n, m1)  # upper bound of support
        x = randint.rvs(xl, xu, size=xl.shape, random_state=rng)
        odds = rng.random(x.shape)*2
        self.x, self.N, self.m1, self.n, self.odds = x, N, m1, n, odds

    # test output is more readable when function names (strings) are passed
    @pytest.mark.parametrize('dist_name',
                             ['nchypergeom_fisher', 'nchypergeom_wallenius'])
    def test_nch_hypergeom(self, dist_name):
        # Both noncentral hypergeometric distributions reduce to the
        # hypergeometric distribution when odds = 1
        dists = {'nchypergeom_fisher': nchypergeom_fisher,
                 'nchypergeom_wallenius': nchypergeom_wallenius}
        dist = dists[dist_name]
        x, N, m1, n = self.x, self.N, self.m1, self.n
        assert_allclose(dist.pmf(x, N, m1, n, odds=1),
                        hypergeom.pmf(x, N, m1, n))

    def test_nchypergeom_fisher_naive(self):
        # test against a very simple implementation
        x, N, m1, n, odds = self.x, self.N, self.m1, self.n, self.odds

        @np.vectorize
        def pmf_mean_var(x, N, m1, n, w):
            # simple implementation of nchypergeom_fisher pmf
            m2 = N - m1
            xl = np.maximum(0, n-m2)
            xu = np.minimum(n, m1)

            def f(x):
                t1 = special_binom(m1, x)
                t2 = special_binom(m2, n - x)
                return t1 * t2 * w**x

            def P(k):
                return sum(f(y)*y**k for y in range(xl, xu + 1))

            P0 = P(0)
            P1 = P(1)
            P2 = P(2)
            pmf = f(x) / P0
            mean = P1 / P0
            var = P2 / P0 - (P1 / P0)**2
            return pmf, mean, var

        pmf, mean, var = pmf_mean_var(x, N, m1, n, odds)
        assert_allclose(nchypergeom_fisher.pmf(x, N, m1, n, odds), pmf)
        assert_allclose(nchypergeom_fisher.stats(N, m1, n, odds, moments='m'),
                        mean)
        assert_allclose(nchypergeom_fisher.stats(N, m1, n, odds, moments='v'),
                        var)

    def test_nchypergeom_wallenius_naive(self):
        # test against a very simple implementation

        rng = np.random.RandomState(2)
        shape = (2, 4, 3)
        max_m = 100
        m1 = rng.randint(1, max_m, size=shape)
        m2 = rng.randint(1, max_m, size=shape)
        N = m1 + m2
        n = randint.rvs(0, N, size=N.shape, random_state=rng)
        xl = np.maximum(0, n-m2)
        xu = np.minimum(n, m1)
        x = randint.rvs(xl, xu, size=xl.shape, random_state=rng)
        w = rng.rand(*x.shape)*2

        def support(N, m1, n, w):
            m2 = N - m1
            xl = np.maximum(0, n-m2)
            xu = np.minimum(n, m1)
            return xl, xu

        @np.vectorize
        def mean(N, m1, n, w):
            m2 = N - m1
            xl, xu = support(N, m1, n, w)

            def fun(u):
                return u/m1 + (1 - (n-u)/m2)**w - 1

            return root_scalar(fun, bracket=(xl, xu)).root

        with warnings.catch_warnings():
            warnings.filterwarnings("ignore", category=RuntimeWarning,
                       message="invalid value encountered in mean")
            assert_allclose(nchypergeom_wallenius.mean(N, m1, n, w),
                            mean(N, m1, n, w), rtol=2e-2)

        @np.vectorize
        def variance(N, m1, n, w):
            m2 = N - m1
            u = mean(N, m1, n, w)
            a = u * (m1 - u)
            b = (n-u)*(u + m2 - n)
            return N*a*b / ((N-1) * (m1*b + m2*a))

        with warnings.catch_warnings():
            warnings.filterwarnings("ignore", category=RuntimeWarning,
                       message="invalid value encountered in mean")
            assert_allclose(
                nchypergeom_wallenius.stats(N, m1, n, w, moments='v'),
                variance(N, m1, n, w),
                rtol=5e-2
            )

        @np.vectorize
        def pmf(x, N, m1, n, w):
            m2 = N - m1
            xl, xu = support(N, m1, n, w)

            def integrand(t):
                D = w*(m1 - x) + (m2 - (n-x))
                res = (1-t**(w/D))**x * (1-t**(1/D))**(n-x)
                return res

            def f(x):
                t1 = special_binom(m1, x)
                t2 = special_binom(m2, n - x)
                the_integral = quad(integrand, 0, 1,
                                    epsrel=1e-16, epsabs=1e-16)
                return t1 * t2 * the_integral[0]

            return f(x)

        pmf0 = pmf(x, N, m1, n, w)
        pmf1 = nchypergeom_wallenius.pmf(x, N, m1, n, w)

        atol, rtol = 1e-6, 1e-6
        i = np.abs(pmf1 - pmf0) < atol + rtol*np.abs(pmf0)
        assert i.sum() > np.prod(shape) / 2  # works at least half the time

        # for those that fail, discredit the naive implementation
        for N, m1, n, w in zip(N[~i], m1[~i], n[~i], w[~i]):
            # get the support
            m2 = N - m1
            xl, xu = support(N, m1, n, w)
            x = np.arange(xl, xu + 1)

            # calculate sum of pmf over the support
            # the naive implementation is very wrong in these cases
            assert pmf(x, N, m1, n, w).sum() < .5
            assert_allclose(nchypergeom_wallenius.pmf(x, N, m1, n, w).sum(), 1)

    def test_wallenius_against_mpmath(self):
        # precompute data with mpmath since naive implementation above
        # is not reliable. See source code in gh-13330.
        M = 50
        n = 30
        N = 20
        odds = 2.25
        # Expected results, computed with mpmath.
        sup = np.arange(21)
        pmf = np.array([3.699003068656875e-20,
                        5.89398584245431e-17,
                        2.1594437742911123e-14,
                        3.221458044649955e-12,
                        2.4658279241205077e-10,
                        1.0965862603981212e-08,
                        3.057890479665704e-07,
                        5.622818831643761e-06,
                        7.056482841531681e-05,
                        0.000618899425358671,
                        0.003854172932571669,
                        0.01720592676256026,
                        0.05528844897093792,
                        0.12772363313574242,
                        0.21065898367825722,
                        0.24465958845359234,
                        0.1955114898110033,
                        0.10355390084949237,
                        0.03414490375225675,
                        0.006231989845775931,
                        0.0004715577304677075])
        mean = 14.808018384813426
        var = 2.6085975877923717

        # nchypergeom_wallenius.pmf returns 0 for pmf(0) and pmf(1), and pmf(2)
        # has only three digits of accuracy (~ 2.1511e-14).
        assert_allclose(nchypergeom_wallenius.pmf(sup, M, n, N, odds), pmf,
                        rtol=1e-13, atol=1e-13)
        assert_allclose(nchypergeom_wallenius.mean(M, n, N, odds),
                        mean, rtol=1e-13)
        assert_allclose(nchypergeom_wallenius.var(M, n, N, odds),
                        var, rtol=1e-11)

    @pytest.mark.parametrize('dist_name',
                             ['nchypergeom_fisher', 'nchypergeom_wallenius'])
    def test_rvs_shape(self, dist_name):
        # Check that when given a size with more dimensions than the
        # dimensions of the broadcast parameters, rvs returns an array
        # with the correct shape.
        dists = {'nchypergeom_fisher': nchypergeom_fisher,
                 'nchypergeom_wallenius': nchypergeom_wallenius}
        dist = dists[dist_name]
        x = dist.rvs(50, 30, [[10], [20]], [0.5, 1.0, 2.0], size=(5, 1, 2, 3))
        assert x.shape == (5, 1, 2, 3)


@pytest.mark.parametrize("mu, q, expected",
                         [[10, 120, -1.240089881791596e-38],
                          [1500, 0, -86.61466680572661]])
def test_nbinom_11465(mu, q, expected):
    # test nbinom.logcdf at extreme tails
    size = 20
    n, p = size, size/(size+mu)
    # In R:
    # options(digits=16)
    # pnbinom(mu=10, size=20, q=120, log.p=TRUE)
    assert_allclose(nbinom.logcdf(q, n, p), expected)


def test_gh_17146():
    # Check that discrete distributions return PMF of zero at non-integral x.
    # See gh-17146.
    x = np.linspace(0, 1, 11)
    p = 0.8
    pmf = bernoulli(p).pmf(x)
    i = (x % 1 == 0)
    assert_allclose(pmf[-1], p)
    assert_allclose(pmf[0], 1-p)
    assert_equal(pmf[~i], 0)


class TestBetaNBinom:
    @pytest.mark.parametrize('x, n, a, b, ref',
                            [[5, 5e6, 5, 20, 1.1520944824139114e-107],
                            [100, 50, 5, 20, 0.002855762954310226],
                            [10000, 1000, 5, 20, 1.9648515726019154e-05]])
    def test_betanbinom_pmf(self, x, n, a, b, ref):
        # test that PMF stays accurate in the distribution tails
        # reference values computed with mpmath
        # from mpmath import mp
        # mp.dps = 500
        # def betanbinom_pmf(k, n, a, b):
        #     k = mp.mpf(k)
        #     a = mp.mpf(a)
        #     b = mp.mpf(b)
        #     n = mp.mpf(n)
        #     return float(mp.binomial(n + k - mp.one, k)
        #                  * mp.beta(a + n, b + k) / mp.beta(a, b))
        assert_allclose(betanbinom.pmf(x, n, a, b), ref, rtol=1e-10)


    @pytest.mark.parametrize('n, a, b, ref',
                            [[10000, 5000, 50, 0.12841520515722202],
                            [10, 9, 9, 7.9224400871459695],
                            [100, 1000, 10, 1.5849602176622748]])
    def test_betanbinom_kurtosis(self, n, a, b, ref):
        # reference values were computed via mpmath
        # from mpmath import mp
        # def kurtosis_betanegbinom(n, a, b):
        #     n = mp.mpf(n)
        #     a = mp.mpf(a)
        #     b = mp.mpf(b)
        #     four = mp.mpf(4.)
        #     mean = n * b / (a - mp.one)
        #     var = (n * b * (n + a - 1.) * (a + b - 1.)
        #            / ((a - 2.) * (a - 1.)**2.))
        #     def f(k):
        #         return (mp.binomial(n + k - mp.one, k)
        #                 * mp.beta(a + n, b + k) / mp.beta(a, b)
        #                 * (k - mean)**four)
        #      fourth_moment = mp.nsum(f, [0, mp.inf])
        #      return float(fourth_moment/var**2 - 3.)
        assert_allclose(betanbinom.stats(n, a, b, moments="k"),
                        ref, rtol=3e-15)


class TestZipf:
    def test_gh20692(self):
        # test that int32 data for k generates same output as double
        k = np.arange(0, 1000)
        k_int32 = k.astype(np.int32)
        dist = zipf(9)
        pmf = dist.pmf(k)
        pmf_k_int32 = dist.pmf(k_int32)
        assert_equal(pmf, pmf_k_int32)


def test_gh20048():
    # gh-20048 reported an infinite loop in _drv2_ppfsingle
    # check that the one identified is resolved
    class test_dist_gen(stats.rv_discrete):
        def _cdf(self, k):
            return min(k / 100, 0.99)

    test_dist = test_dist_gen(b=np.inf)

    message = "Arguments that bracket..."
    with pytest.raises(RuntimeError, match=message):
        test_dist.ppf(0.999)


class TestPoissonBinomial:
    def test_pmf(self):
        # Test pmf against R `poisbinom` to confirm that this is indeed the Poisson
        # binomial distribution. Consistency of other methods and all other behavior
        # should be covered by generic tests. (If not, please add a generic test.)
        # Like many other distributions, no special attempt is made to be more
        # accurate than the usual formulas provide, so we use default tolerances.
        #
        # library(poisbinom)
        # options(digits=16)
        # k = c(0, 1, 2, 3, 4)
        # p = c(0.9480654803913988, 0.052428488100509374,
        #       0.25863527358887417, 0.057764076043633206)
        # dpoisbinom(k, p)
        rng = np.random.default_rng(259823598254)
        n = rng.integers(10)  # 4
        k = np.arange(n + 1)
        p = rng.random(n)  #  [0.9480654803913988, 0.052428488100509374,
                           #   0.25863527358887417, 0.057764076043633206]
        res = poisson_binom.pmf(k, p)
        ref = [0.0343763443678060318, 0.6435428452689714307, 0.2936345519235536994,
               0.0277036647503902354, 0.0007425936892786034]
        assert_allclose(res, ref)


class TestRandInt:
    def test_gh19759(self):
        # test zero PMF values within the support reported by gh-19759
        a = -354
        max_range = abs(a)
        all_b_1 = [a + 2 ** 31 + i for i in range(max_range)]
        res = randint.pmf(325, a, all_b_1)
        assert (res > 0).all()
        ref = 1 / (np.asarray(all_b_1, dtype=np.float64) - a)
        assert_allclose(res, ref)