Add the Irwin-Hall distribution

Appease linter for line length

scipy/stats/__init__.py

@@ -97,6 +97,7 @@
    invgamma          -- Inverse Gamma
    invgauss          -- Inverse Gaussian
    invweibull        -- Inverse Weibull
+   irwinhall         -- Irwin-Hall
    jf_skew_t         -- Jones and Faddy Skew-T
    johnsonsb         -- Johnson SB
    johnsonsu         -- Johnson SU

scipy/stats/_continuous_distns.py

@@ -9,6 +9,7 @@
 
 import numpy as np
 from numpy.polynomial import Polynomial
+from scipy.interpolate import BSpline
 from scipy._lib.doccer import (extend_notes_in_docstring,
                                replace_notes_in_docstring,
                                inherit_docstring_from)
@@ -24,7 +25,7 @@
 from ._tukeylambda_stats import (tukeylambda_variance as _tlvar,
                                  tukeylambda_kurtosis as _tlkurt)
 from ._distn_infrastructure import (
-    get_distribution_names, _kurtosis,
+    get_distribution_names, _kurtosis, _isintegral,
     rv_continuous, _skew, _get_fixed_fit_value, _check_shape, _ShapeInfo)
 from ._ksstats import kolmogn, kolmognp, kolmogni
 from ._constants import (_XMIN, _LOGXMIN, _EULER, _ZETA3, _SQRT_PI,
@@ -9033,6 +9034,87 @@ def _munp(self, n, b):
 
 rice = rice_gen(a=0.0, name="rice")
 
+class irwinhall_gen(rv_continuous):
+    r"""
+    An Irwin-Hall continuous random variable is the sum of :math`n` independent
+    standard uniform r.v.'s. 
+    
+    %(before_notes)s
+
+    Notes
+    -----
+    Applications include Rao's Spacing Test, 
+    a more powerful alternative to the Rayleigh test 
+    when the data are not unimodal, and radar.
+
+    Conveniently, the pdf and cdf are the :math`n`-fold convolution of 
+    the ones for the standard uniform distribution,
+    which are well known to be just cardinal B-splines of degree :math`n-1` 
+    with knots :math`\left{0,1,\dots,n-1\right}`.
+    %(after_notes)s
+
+    References
+    ----------
+    ..  [1] "Irwin-Hall distribution",
+            https://en.wikipedia.org/wiki/Irwin-Hall_distribution
+        [2] "Rao's Spacing Test",
+            https://jammalam.faculty.pstat.ucsb.edu/html/favorite/test.htm
+        [3] HALL PHILIP, "THE DISTRIBUTION OF MEANS FOR SAMPLES OF SIZE N DRAWN 
+            FROM A POPULATION IN WHICH THE VARIATE TAKES VALUES BETWEEN 0 AND 1, 
+            ALL SUCH VALUES BEING EQUALLY PROBABLE", 
+            Biometrika, Volume 19, Issue 3-4, December 1927, Pages 240-244, 
+            https://doi.org/10.1093/biomet/19.3-4.240
+        [4] J. O. IRWIN, "ON THE FREQUENCY DISTRIBUTION OF THE MEANS OF SAMPLES 
+            FROM A POPULATION HAVING ANY LAW OF FREQUENCY WITH FINITE MOMENTS, 
+            WITH SPECIAL REFERENCE TO PEARSON'S TYPE II", 
+            Biometrika, Volume 19, Issue 3-4, December 1927, Pages 225-239, 
+            https://doi.org/10.1093/biomet/19.3-4.225
+        [5] K. Buchanan, T. Adeyemi, C. Flores-Molina, S. Wheeland and D. Overturf, 
+            "Sidelobe behavior and bandwidth characteristics 
+            of distributed antenna arrays," 
+            2018 United States National Committee of 
+            URSI National Radio Science Meeting (USNC-URSI NRSM), 
+            Boulder, CO, USA, 2018, pp. 1-2.
+            https://www.usnc-ursi-archive.org/nrsm/2018/papers/B15-9.pdf
+
+    %(example)s
+
+    from scipy.stats import irwinhall
+
+    bates = irwinhall(n, name="bates", scale=1/n)
+    """
+
+    def _argcheck(self, n):
+        return (n > 0) & _isintegral(n)
+
+    def _get_support(self, n):
+        return 0, n 
+
+    def _shape_info(self):
+        return [_ShapeInfo("n", True, (1, np.inf), (True, False))]
+
+    def _munp(self, n, t):
+        return (sc.expm1(t)/t)**n
+
+    @staticmethod
+    def _cardinal_bspline(n):
+        t = np.arange(n+1) 
+        return BSpline.basis_element(t, extrapolate=False)
+
+    def _pdf(self, x, n):
+        return self._cardinal_bspline(n)(x)
+
+    def _cdf(self, x, n):
+        cdf = self._cardinal_bspline(n).antiderivative()
+        return cdf(x)
+
+    def _stats(self, n):
+        # mgf = ((exp(t) - 1)/t)**n
+        # m'th derivative follows from the generalized Leibniz rule
+
+        return n/2, n/12, 0,  3-6/(5*n)
+
+irwinhall = irwinhall_gen(name="irwinhall")    
 
 class recipinvgauss_gen(rv_continuous):
     r"""A reciprocal inverse Gaussian continuous random variable.

scipy/stats/_discrete_distns.py

@@ -14,16 +14,13 @@
 import numpy as np
 
 from ._distn_infrastructure import (rv_discrete, get_distribution_names,
-                                    _check_shape, _ShapeInfo)
+                                    _check_shape, _ShapeInfo, _isintegral)
 from ._biasedurn import (_PyFishersNCHypergeometric,
                          _PyWalleniusNCHypergeometric,
                          _PyStochasticLib3)
 import scipy.special._ufuncs as scu
 
 
-def _isintegral(x):
-    return x == np.round(x)
-
 
 class binom_gen(rv_discrete):
     r"""A binomial discrete random variable.

scipy/stats/_distn_infrastructure.py

@@ -430,6 +430,8 @@ def _fit_determine_optimizer(optimizer):
             raise ValueError("%s is not a valid optimizer" % optimizer) from e
     return optimizer
 
+def _isintegral(x):
+    return x == np.round(x)
 
 def _sum_finite(x):
     """

scipy/stats/_distr_params.py

@@ -55,6 +55,7 @@
     ['invgamma', (4.0668996136993067,)],
     ['invgauss', (0.14546264555347513,)],
     ['invweibull', (10.58,)],
+    ['irwinhall', (10)],
     ['jf_skew_t', (8, 4)],
     ['johnsonsb', (4.3172675099141058, 3.1837781130785063)],
     ['johnsonsu', (2.554395574161155, 2.2482281679651965)],
@@ -226,6 +227,7 @@
     ['invgamma', (-1, )],
     ['invgauss', (-1, )],
     ['invweibull', (-1, )],
+    ['irwinhall', (-1, 0, 0.5)],
     ['jf_skew_t', (-1, 0)],
     ['johnsonsb', (1, -2)],
     ['johnsonsu', (1, -2)],