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[[File:KS Example.png|thumb|300px|Illustration of the Kolmogorov-Smirnov statistic. Red line is CDF, blue line is an [[ECDF]], and the black arrow is the K-S statistic.]]

In [[statistics]], the '''[[Andrey Kolmogorov|Kolmogorov]]–[[Nikolai Smirnov (mathematician)|Smirnov]] test (K–S test)''' is a [[nonparametric statistics|nonparametric test]] for the equality of continuous, one-dimensional [[probability distribution]]s that can be used to compare a [[random sample|sample]] with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). The Kolmogorov–Smirnov statistic quantifies a [[metric (mathematics)|distance]] between the [[empirical distribution function]] of the sample and the [[cumulative distribution function]] of the reference distribution, or between the empirical distribution functions of two samples. The [[null distribution]] of this statistic is calculated under the [[null hypothesis]] that the samples are drawn from the same distribution (in the two-sample case) or that the sample is drawn from the reference distribution (in the one-sample case). In each case, the distributions considered under the null hypothesis are continuous distributions but are otherwise unrestricted.

The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.

The Kolmogorov–Smirnov test can be modified to serve as a [[goodness of fit]] test. In the special case of testing for [[Normal distribution|normality]] of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic: see [[#Test with estimated parameters|below]]. Various studies have found that, even in this corrected form, the test is less powerful for testing normality than the [[Shapiro–Wilk test]] or [[Anderson–Darling test]].<ref>{{cite journal
 | first = M. A. | last = Stephens | coauthors =
 | year = 1974 | month = 
 | title = EDF Statistics for Goodness of Fit and Some Comparisons
 | journal = Journal of the American Statistical Association
 | volume = 69 | issue = 347| pages = 730–737 | jstor =2286009
 | doi = 10.2307/2286009
 | publisher = American Statistical Association
 }}</ref> However, other test have their own disadvantages. For instance the Shapiro-Wilk test is known not to work well with many ties (many identical values).

==Kolmogorov–Smirnov statistic==
The [[empirical distribution function]] ''F''<sub>''n''</sub> for ''n'' [[Independent and identically distributed random variables|iid]] observations ''X<sub>i</sub>'' is defined as

:<math>F_n(x)={1 \over n}\sum_{i=1}^n I_{X_i\leq x}</math>

where <math>I_{X_i\leq x}</math> is the [[indicator function]], equal to 1 if ''X''<sub>''i''</sub>&nbsp;≤&nbsp;''x'' and equal to 0 otherwise.

The '''Kolmogorov–Smirnov [[statistic]]''' for a given [[cumulative distribution function]] ''F''(''x'') is

:<math>D_n=\sup_x |F_n(x)-F(x)|</math>

where ''sup&nbsp;<sub>x''</sub> is the [[supremum]] of the set of distances. By the [[Glivenko–Cantelli theorem]], if the sample comes from distribution ''F''(''x''), then ''D''<sub>''n''</sub> converges to 0 [[almost surely]] in the limit when <math>n</math> goes to infinity. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see below). [[Donsker's theorem]] provides yet a stronger result.

In practice, the statistic requires a relatively large number of data points to properly reject the null hypothesis.

==Kolmogorov distribution==
The '''Kolmogorov distribution''' is the distribution of the [[random variable]]

:<math>K=\sup_{t\in[0,1]}|B(t)|</math>

where ''B''(''t'') is the [[Brownian bridge]]. The [[cumulative distribution function]] of ''K'' is given by<ref>{{Cite journal |author=Marsaglia G, Tsang WW, Wang J |year=2003 |title=Evaluating Kolmogorov’s Distribution |journal=Journal of Statistical Software |volume=8 |issue=18 |pages=1–4 |url=http://www.jstatsoft.org/v08/i18/paper}}</ref>

:<math>\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)}.</math>

Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by [[Andrey Kolmogorov]],<ref name=AK>{{Cite journal |author=Kolmogorov A |year=1933 |title=Sulla determinazione empirica di una legge di distribuzione |journal=G. Ist. Ital. Attuari |volume=4 |pages=83–91}}</ref> while a table of the distribution was published by Nikolai Vasilyevich Smirnov.<ref>{{Cite journal |author=Smirnov N |year=1948 |title=Table for estimating the goodness of fit of empirical distributions |journal=[[Annals of Mathematical Statistics]] |volume=19 |pages=279–281}}</ref> Recurrence relations for the distribution of the test statistic in finite samples are available.<ref name=AK/>

Under null hypothesis that the sample comes from the hypothesized distribution ''F''(''x''),

:<math>\sqrt{n}D_n\xrightarrow{n\to\infty}\sup_t |B(F(t))|</math>

[[convergence of random variables|in distribution]], where ''B''(''t'') is the [[Brownian bridge]].

If ''F'' is continuous then under the null hypothesis <math>\sqrt{n}D_n</math> converges to the Kolmogorov distribution, which does not depend on ''F''. This result may also be known as the '''Kolmogorov theorem'''; see [[Kolmogorov's theorem]] for disambiguation.

The ''goodness-of-fit'' test or the Kolmogorov–Smirnov test is constructed by using the critical values of the Kolmogorov distribution. The null hypothesis is rejected at level <math>\alpha</math> if

:<math>\sqrt{n}D_n>K_\alpha,\,</math>

where ''K''<sub>''α''</sub> is found from

:<math>\operatorname{Pr}(K\leq K_\alpha)=1-\alpha.\,</math>

The asymptotic [[statistical power|power]] of this test is 1.

===Test with estimated parameters===

If either the form or the parameters of ''F''(''x'') are determined from the data ''X''<sub>''i''</sub> the critical values determined in this way are invalid. In such cases, [[Monte Carlo method|Monte Carlo]] or other methods may be required, but tables have been prepared for some cases. Details for the required modifications to the test statistic and for the critical values for  the [[normal distribution]] and the [[exponential distribution]] have been published,<ref name="Pearson & Hartley">{{cite book
|title= Biometrika Tables for Statisticians |editors = Pearson E.S. and Hartley, H.O. |year= 1972 |volume= 2 |publisher= Cambridge University Press |ISBN= 0-521-06937-8|pages=117–123, Tables 54, 55}}</ref> and later publications also include the [[Gumbel distribution]].<ref name="Shorak & Wellner">{{cite book |title= Empirical Processes with Applications to Statistics |authors= Galen R. Shorack and Jon A. Wellner |year= 1986 |ISBN= 047186725X |publisher= Wiley |page= 239}}</ref> The [[Lilliefors test]] represents a special case of this for the normal distribution.
The logarithm transformation may help to overcome cases were the Kolmogorov test data does not seem to fit the assumption that it came from the normal distribution.

==Two-sample Kolmogorov–Smirnov test==
[[File:KS2 Example.png|thumb|300px|Illustration of the Two-Sample Kolmogorov-Smirnov statistic. Red and blue lines each correspond to an empirical distribution function, and the black arrow is the two-sample K-S statistic.]]

The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the Kolmogorov–Smirnov statistic is

:<math>D_{n,n'}=\sup_x |F_{1,n}(x)-F_{2,n'}(x)|,</math>

where <math>F_{1,n}</math> and <math>F_{2,n'}</math> are the [[empirical distribution function]]s of the first and the second sample respectively.

The null hypothesis is rejected at level <math>\alpha</math> if

:<math>D_{n,n'}>c(\alpha)\sqrt{\frac{n + n'}{n n'}}.</math><ref name="TableTwoSample">[http://www.soest.hawaii.edu/wessel/courses/gg313/Critical_KS.pdf Table of critical values for the two-sample test]</ref>

The value of <math>c({\alpha})</math> is given in the table below for each level of <math>\alpha</math><ref name="TableTwoSample"/>

{| class="wikitable"
|-
| <math>\alpha</math> || 0.10 || 0.05 || 0.025 || 0.01 || 0.005 || 0.001
|-
| <math>c({\alpha})</math> || 1.22 || 1.36 || 1.48 || 1.63 || 1.73 || 1.95
|}

Note that the two-sample test checks whether the two data samples come from the same distribution.  This does not specify what that common
distribution is (e.g. normal or not normal). Again, tables of critical values have been published.<ref name="Pearson & Hartley"/><ref name="TableTwoSample"/>

==Setting confidence limits for the shape of a distribution function==

While the Kolmogorov–Smirnov test is usually used to test whether a given ''F''(''x'') is the underlying probability distribution of ''F''<sub>''n''</sub>(''x''), the procedure may be inverted to give confidence limits on ''F''(''x'') itself. If one chooses a critical value of the test statistic ''D''<sub>''α''</sub> such that P(''D''<sub>''n''</sub>&nbsp;>&nbsp;''D''<sub>''α''</sub>) = ''α'', then a band of width ±''D''<sub>''α''</sub> around ''F''<sub>''n''</sub>(''x'') will entirely contain ''F''(''x'') with probability 1&nbsp;−&nbsp;''α''.

==The Kolmogorov–Smirnov statistic in more than one dimension==

A distribution free multivariate Kolmogorov-Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997).<ref>{{cite journal |last=Justel |first=A. |last2=Peña |first2=D. |last3=Zamar |first3=R. |year=1997 |title=A multivariate Kolmogorov-Smirnov test of goodness of fit |journal=Statistics & Probability Letters |volume=35 |issue=3 |pages=251–259 |doi=10.1016/S0167-7152(97)00020-5 }}</ref>  The test uses a statistic which is built using Rosenblatt's transformation and an algorithm is developed to compute it in the bivariate case.  An approximate test, that can be easily computed in any dimension, is also presented.

The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to [[multivariate statistics|multivariate data]]. This is not straightforward because the maximum difference between two joint [[cumulative distribution function]]s is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of <math>\Pr(x < X \and y < Y)</math> or <math>\Pr(X < x \and Y > y)</math> or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.

One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K-S statistics.  In ''d'' dimensions, there are 2<sup>''d''</sup>−1 such orderings.  One such variation is due to Peacock<ref name="Peacock">{{cite journal  |author = Peacock J.A. |title = Two-dimensional goodness-of-fit testing in astronomy |journal = Monthly Notices of the Royal Astronomical Society |volume = 202 |pages = 615–627  |year = 1983 |bibcode = 1983MNRAS.202..615P }}</ref> and another to Fasano and Franceschini<ref name="Fasano">{{cite journal |authors= Fasano, G., Franceschini, A. |year=1987 |title= A multidimensional version of the Kolmogorov–Smirnov test |journal= Monthly Notices of the Royal Astronomical Society |issn=0035-8711 |volume= 225 |pages= 155–170 |bibcode=1987MNRAS.225..155F }}</ref> (see Lopes et al. for a comparison and computational details).<ref name="Lopes">{{cite conference |authors= Lopes, R.H.C., Reid, I., Hobson, P.R. |year=2007 |title= The two-dimensional Kolmogorov-Smirnov test |conference= XI International Workshop on Advanced Computing and Analysis Techniques in Physics Research |date= April 23–27, 2007 |location= Amsterdam, the Netherlands |url= http://dspace.brunel.ac.uk/bitstream/2438/1166/1/acat2007.pdf }}</ref> Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.

==See also==

* [[Cramér–von Mises criterion|Cramér–von Mises test]]
* [[Dvoretzky–Kiefer–Wolfowitz inequality]]
* [[Jarque–Bera test]]
* [[Kuiper's test]]
* [[P-value]]
* [[Siegel–Tukey test]]
* [[Total variation distance of probability measures]]

==Footnotes==
{{Reflist}}

==References==
* {{cite book
  | last = Eadie
  | first = W.T.
  | coauthors = D. Drijard, F.E. James, M. Roos and B. Sadoulet
  | title = Statistical Methods in Experimental Physics
  | publisher = North-Holland
  | year = 1971
  | location = Amsterdam
  | pages = 269–271
  | isbn = 0-444-10117-9 }}
* {{cite book
  | last1 = Stuart
  | first1 = Alan
  | first2 = Keith
  | last2 = Ord
  | first3=Steven [F.]
  | last3=Arnold
  | title=Classical Inference and the Linear Model
  | edition=Sixth
  | series = Kendall's Advanced Theory of Statistics
  | volume = 2A
  | year = 1999
  | publisher = Arnold
  | location = London
  | isbn=0-340-66230-1
  | mr=1687411
  | pages = 25.37–25.43 }}
*{{cite book |last=Corder |first=G. W. |last2=Foreman |first2=D. I. |year=2009 |title=Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach |location= |publisher=Wiley |isbn=978-0-470-45461-9 }}
*{{cite journal |last=Stephens |first=M. A. |year=1979 |title=Test of fit for the logistic distribution based on the empirical distribution function |journal=Biometrika |volume=66 |issue=3 |pages=591–595 |doi=10.1093/biomet/66.3.591 }}

==External links==
*{{springer|title=Kolmogorov-Smirnov test|id=p/k055740}}
*[http://www.physics.csbsju.edu/stats/KS-test.html Short introduction]
*[http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm KS test explanation]
*[http://www.ciphersbyritter.com/JAVASCRP/NORMCHIK.HTM JavaScript implementation of one- and two-sided tests]
*[http://jumk.de/statistic-calculator/ Online calculator with the K-S test]
* Open-source C++ code to compute the [http://root.cern.ch/root/html/TMath.html#TMath:KolmogorovProb Kolmogorov distribution] and perform the [http://root.cern.ch/root/html/TMath.html#TMath:KolmogorovTest K-S test]
*Paper on [http://www.jstatsoft.org/v08/i18/paper Evaluating Kolmogorov’s Distribution]; contains C implementation. This is the method used in [[Matlab]].

{{Statistics}}

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[[Category:Statistical tests]]
[[Category:Non-parametric statistics]]
[[Category:Normality tests]]