# Difference between revisions of "Kolmogorov-Smirnov test"

2010 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  showed that (1) 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  that if and , then (2)

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.

How to Cite This Entry:
Kolmogorov-Smirnov test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov-Smirnov_test&oldid=21350
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article