Kolmogorov-Smirnov test

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2020 Mathematics Subject Classification: Primary: 62G10 [MSN][ZBL]

A non-parametric test used for testing a hypothesis , according to which independent random variables have a given continuous distribution function , against the one-sided alternative : , where is the mathematical expectation of the empirical distribution function . The Kolmogorov–Smirnov test is constructed from the statistic

where is the variational series (or set of order statistics) obtained from the sample . Thus, the Kolmogorov–Smirnov test is a variant of the Kolmogorov test for testing the hypothesis against a one-sided alternative . By studying the distribution of the statistic , N.V. Smirnov [1] showed that


where and is the integer part of the number . Smirnov obtained in addition to the exact distribution (1) of its limit distribution, namely: If and , then

where is any positive number. By means of the technique of asymptotic Pearson transformation it has been proved [2] that if and , then


According to the Kolmogorov–Smirnov test, the hypothesis must be rejected with significance level whenever

where, by virtue of (2),

The testing of against the alternative : is dealt with similarly. In this case the statistic of the Kolmogorov–Smirnov test is the random variable

whose distribution is the same as that of the statistic when is true.


[1] N.V. Smirnov, "Approximate distribution laws for random variables, constructed from empirical data" Uspekhi Mat. Nauk , 10 (1944) pp. 179–206 (In Russian)
[2] L.N. Bol'shev, "Asymptotically Pearson transformations" Theor. Probab. Appl. , 8 (1963) pp. 121–146 Teor. Veroyatnost. i Primenen. , 8 : 2 (1963) pp. 129–155
[3] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[4] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)


There is also a two-sample Kolmogorov–Smirnov test, cf. the editorial comments to Kolmogorov test and, for details, [a1], [a2].


[a1] G.E. Noether, "A brief survey of nonparametric statistics" R.V. Hogg (ed.) , Studies in statistics , Math. Assoc. Amer. (1978) pp. 39–65
[a2] M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973)
How to Cite This Entry:
Kolmogorov-Smirnov test. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article