View source for Kolmogorov-Smirnov test

{{MSC|62G10}}

[[Category:Nonparametric inference]]

A [[Non-parametric test|non-parametric test]] used for testing a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557401.png" />, according to which independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557402.png" /> have a given continuous distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557403.png" />, against the one-sided alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557404.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557405.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557406.png" /> is the mathematical expectation of the empirical distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557407.png" />. The Kolmogorov–Smirnov test is constructed from the statistic

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557408.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k0557409.png" /> is the [[Variational series|variational series]] (or set of order statistics) obtained from the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574010.png" />. Thus, the Kolmogorov–Smirnov test is a variant of the [[Kolmogorov test|Kolmogorov test]] for testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574011.png" /> against a one-sided alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574012.png" />. By studying the distribution of the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574013.png" />, N.V. Smirnov [[#References|[1]]] showed that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574017.png" /> is the integer part of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574018.png" />. Smirnov obtained in addition to the exact distribution (1) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574019.png" /> its limit distribution, namely: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574021.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574023.png" /> is any positive number. By means of the technique of asymptotic Pearson transformation it has been proved [[#References|[2]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574025.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

According to the Kolmogorov–Smirnov test, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574027.png" /> must be rejected with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574028.png" /> whenever

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574029.png" /></td> </tr></table>

where, by virtue of (2),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574030.png" /></td> </tr></table>

The testing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574031.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574032.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574033.png" /> is dealt with similarly. In this case the statistic of the Kolmogorov–Smirnov test is the random variable

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574034.png" /></td> </tr></table>

whose distribution is the same as that of the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574035.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055740/k05574036.png" /> is true.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Smirnov,   "Approximate distribution laws for random variables, constructed from empirical data"  ''Uspekhi Mat. Nauk'' , '''10'''  (1944)  pp. 179–206  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR></table>



====Comments====
There is also a two-sample Kolmogorov–Smirnov test, cf. the editorial comments to [[Kolmogorov test|Kolmogorov test]] and, for details, [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Noether,   "A brief survey of nonparametric statistics"  R.V. Hogg (ed.) , ''Studies in statistics'' , Math. Assoc. Amer.  (1978)  pp. 39–65</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hollander,   D.A. Wolfe,   "Nonparametric statistical methods" , Wiley  (1973)</TD></TR></table>