# 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 $$\sqrt{\frac{mn}{m+n}}D_{m,n}$$ values, as $$m,n\rightarrow\infty$$ and $$m/n\rightarrow a \ne 0, \infty$$.

There are no shape parameters, and the support is $$x\in\left[0,\infty\right)$$.

\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*}

## 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