doc/source/tutorial/stats/continuous_ksone.rst
.. _continuous-ksone:
This is the distribution of maximum positive differences between an
empirical distribution function, computed from :math:n samples or observations,
and a comparison (or target) cumulative distribution function.
Writing :math:D_n^+ = \sup_t \left(F_{empirical,n}(t)-F_{target}(t)\right),
ksone is the distribution of the :math:D_n^+ values.
(The distribution of :math:D_n^- = \sup_t \left(F_{target}(t)-F_{empirical,n}(t)\right)
differences follows the same distribution, so ksone can be used for one-sided tests on either side.)
There is one shape parameter :math:n, a positive integer, and the support is :math:x\in\left[0,1\right].
.. math:: :nowrap:
\begin{eqnarray*} F\left(n, x\right) & = & 1 - \sum_{j=0}^{\lfloor n(1-x)\rfloor} \dbinom{n}{j} x \left(x+\frac{j}{n}\right)^{j-1} \left(1-x-\frac{j}{n}\right)^{n-j}\\
& = & 1 - \textrm{scipy.special.smirnov}(n, x) \\
\lim_{n \rightarrow\infty} F\left(n, \frac{x}{\sqrt n}\right) & = & e^{-2 x^2} \end{eqnarray*}
"Kolmogorov-Smirnov test", Wikipedia https://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test
Birnbaum, Z. W.; Tingey, Fred H. "One-Sided Confidence Contours for Probability Distribution Functions." Ann. Math. Statist. 22 (1951), no. 4, 592--596.
Implementation: scipy.stats.ksone