Kolmogorov test

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

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 [BS] 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 [B], 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.).

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.

References

[K]	A.N. Kolmogorov, "Sulla determinizione empirica di una legge di distribuzione" Giorn. Ist. Ital. Attuari , 4 (1933) pp. 83–91
[S]	N.V. Smirnov, "On estimating the discrepancy between empirical distribiution curves for two independent samples" Byull. Moskov. Gos. Univ. Ser. A , 2 : 2 (1938) pp. 3–14 (In Russian)
[B]	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 Zbl 0125.09103
[BS]	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) Zbl 0529.62099

Comments

Tests based on and , and similar tests for a two-sample problem based on and , where is the empirical distribution function for samples of size for a population with distribution function , are also called Kolmogorov–Smirnov tests, cf. also Kolmogorov–Smirnov test.

References

[N]	G.E. Noether, "A brief survey of nonparametric statistics" R.V. Hogg (ed.) , Studies in statistics , Math. Assoc. Amer. (1978) pp. 3–65 Zbl 0413.62023
[HW]	M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973) MR0353556 Zbl 0277.62030