# 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 [1] 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 [2] 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.

#### References

 [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)