doc/source/tutorial/stats/continuous_kstwobign.rst
.. _continuous-kstwobign:
This is the limiting distribution of the normalized maximum absolute differences between an
empirical distribution function, computed from :math:n samples or observations,
and a comparison (or target) cumulative distribution function. (ksone is the distribution
of the unnormalized positive differences, :math:D_n^+.)
Writing :math:D_n = \sup_t \left|F_{empirical,n}(t) - F_{target}(t)\right|,
the normalization factor is :math:\sqrt{n}, and kstwobign is the limiting distribution
of the :math:\sqrt{n} D_n values as :math:n\rightarrow\infty.
Note that :math:D_n=\max(D_n^+, D_n^-), but :math:D_n^+ and :math:D_n^- are not independent.
kstwobign can also be used with the differences between two empirical distribution functions,
for sets of observations with :math:m and :math:n samples respectively,
where :math:m and :math:n are "big".
Writing :math:D_{m,n} = \sup_t \left|F_{1,m}(t)-F_{2,n}(t)\right|, where
:math:F_{1,m} and :math:F_{2,n} are the two empirical distribution functions, then
kstwobign is also the limiting distribution of the :math:\sqrt{\frac{mn}{m+n}}D_{m,n} values,
as :math:m,n\rightarrow\infty and :math:m/n\rightarrow a \ne 0, \infty.
There are no shape parameters, and the support is :math:x\in\left[0,\infty\right).
.. math:: :nowrap:
\begin{eqnarray*} F\left(x\right) & = & 1 - 2 \sum_{k=1}^{\infty} (-1)^{k-1} e^{-2k^2 x^2}\\
& = & \frac{\sqrt{2\pi}}{x} \sum_{k=1}^{\infty} e^{-(2k-1)^2 \pi^2/(8x^2)}\\
& = & 1 - \textrm{scipy.special.kolmogorov}(n, x) \\
f\left(x\right) & = & 8x \sum_{k=1}^{\infty} (-1)^{k-1} k^2 e^{-2k^2 x^2} \end{eqnarray*}
"Kolmogorov-Smirnov test", Wikipedia https://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test
Kolmogoroff, A. "Confidence Limits for an Unknown Distribution Function."" Ann. Math. Statist. 12 (1941), no. 4, 461--463.
Smirnov, N. "On the estimation of the discrepancy between empirical curves of distribution for two independent samples" Bull. Math. Univ. Moscou., 2 (1039), 2-26.
Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical Distributions." Ann. Math. Statist. 19 (1948), no. 2, 177--189. and "Errata" Ann. Math. Statist. 21 (1950), no. 2, 301--302.
Implementation: scipy.stats.kstwobign