KStwobign Distribution¶
This is the limiting distribution of the normalized maximum absolute differences between an
empirical distribution function, computed from \(n\) samples or observations,
and a comparison (or target) cumulative distribution function. (ksone
is the distribution
of the unnormalized positive differences, \(D_n^+\).)
Writing \(D_n = \sup_t \left|F_{empirical,n}(t) - F_{target}(t)\right|\),
the normalization factor is \(\sqrt{n}\), and kstwobign
is the limiting distribution
of the \(\sqrt{n} D_n\) values as \(n\rightarrow\infty\).
Note that \(D_n=\max(D_n^+, D_n^-)\), but \(D_n^+\) and \(D_n^-\) are not independent.
kstwobign
can also be used with the differences between two empirical distribution functions,
for sets of observations with \(m\) and \(n\) samples respectively,
where \(m\) and \(n\) are “big”.
Writing \(D_{m,n} = \sup_t \left|F_{1,m}(t)-F_{2,n}(t)\right|\), where
\(F_{1,m}\) and \(F_{2,n}\) are the two empirical distribution functions, then
kstwobign
is also the limiting distribution of the \(\sqrt{\left(\frac{mn}{m+n}\right)D_{m,n}}\) values,
as \(m,n\rightarrow\infty\) and \(m/n\rightarrow a \ne 0, \infty\).
There are no shape parameters, and the support is \(x\in\left[0,\infty\right)\).
References¶
“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