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KStwobign Distribution

docs/tutorial:stats:continuous_kstwobign

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 $D_{n} = sup_{t} ∣ F_{e m p i r i c a l, n} (t) - F_{t a r g e t} (t) ∣$ , the normalization factor is $n$ , and kstwobign is the limiting distribution of the $n D_{n}$ values as $n \to \infty$ .

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 $D_{m, n} = sup_{t} ∣ F_{1, m} (t) - F_{2, n} (t) ∣$ , where $F_{1, m}$ and $F_{2, n}$ are the two empirical distribution functions, then kstwobign is also the limiting distribution of the $\frac{mn}{m + n} D_{m, n}$ values, as $m, n \to \infty$ and $m / n \to a \neq = 0, \infty$ .

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

\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