# Kolmogorov test

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A statistical test used for testing a simple non-parametric hypothesis , according to which independent identically-distributed random variables have a given distribution function , where the alternative hypothesis is taken to be two-sided: where is the mathematical expectation of the empirical distribution function . The critical set of the Kolmogorov test is expressed by the inequality and is based on the following theorem, proved by A.N. Kolmogorov in 1933: If the hypothesis is true, then the distribution of the statistic does not depend on ; also, as , where In 1948 N.V. Smirnov  tabulated the Kolmogorov distribution function . According to the Kolmogorov test with significance level , , the hypothesis must be rejected if , where is the critical value of the Kolmogorov test corresponding to the given significance level and is the root of the equation .

To determine one recommends the use of the approximation of the limiting law of the Kolmogorov statistic and its limiting distribution; see , where it is shown that, as and , (*) The application of the approximation (*) gives the following approximation of the critical value: where is the root of the equation .

In practice, for the calculation of the value of the statistic one uses the fact that where  and is the variational series (or set of order statistics) constructed from the sample . The Kolmogorov test has the following geometric interpretation (see Fig.). Figure: k055760a

The graph of the functions , is depicted in the -plane. The shaded region is the confidence zone at level for the distribution function , since if the hypothesis is true, then according to Kolmogorov's theorem If the graph of does not leave the shaded region then, according to the Kolmogorov test, must be accepted with significance level ; otherwise is rejected.

The Kolmogorov test gave a strong impetus to the development of mathematical statistics, being the start of much research on new methods of statistical analysis lying at the foundations of non-parametric statistics.

How to Cite This Entry:
Kolmogorov test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov_test&oldid=11325
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article