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:
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where is the mathematical expectation of the empirical distribution function
. The critical set of the Kolmogorov test is expressed by the inequality
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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
,
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where
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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
,
![]() | (*) |
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The application of the approximation (*) gives the following approximation of the critical value:
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where is the root of the equation
.
In practice, for the calculation of the value of the statistic one uses the fact that
![]() |
where
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and is the variational series (or set of order statistics) constructed from the sample
. The Kolmogorov test has the following geometric interpretation (see Fig.).
![](/images/math/6/b/3/6b3628c67f85c862c5de59f88784d605.png)
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 |
Kolmogorov test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov_test&oldid=26541