# Kolmogorov-Smirnov test

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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.