# 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 \(x\in\left[0,\infty\right)\).

\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`