Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (K–S test) is a form of minimum distance estimation used as a nonparametric test of equality of one-dimensional probability distributions used to compare a 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 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 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 the sample to modify the null hypothesis reduces the power of a test. Correcting for this bias leads to the Lilliefors test. However, even Lilliefors' modification is less powerful for testing normality than the Shapiro–Wilk test or Anderson–Darling test.^[1]

Contents 1 Kolmogorov–Smirnov statistic 2 Kolmogorov distribution 3 Kolmogorov–Smirnov test 4 Two-sample Kolmogorov–Smirnov test 5 Setting confidence limits for the shape of a distribution function 6 The Kolmogorov–Smirnov statistic in more than one dimension 7 See also 8 References 9 References 10 External links

Kolmogorov–Smirnov statistic

The empirical distribution function F_n for n iid observations X_i is defined as

where is the indicator function, equal to 1 if X_i ≤ x and equal to 0 otherwise.

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

where sup S is the supremum of set S. By the Glivenko–Cantelli theorem, if the sample comes from distribution F(x), then D_n converges to 0 almost surely. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see below). The Donsker theorem provides yet a stronger result.

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

Kolmogorov distribution

The Kolmogorov distribution is the distribution of the random variable

where B(t) is the Brownian bridge. The cumulative distribution function of K is given by

Kolmogorov–Smirnov test

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

in distribution, where B(t) is the Brownian bridge.

If F is continuous then under the null hypothesis 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 α if

where K_α is found from

The asymptotic power of this test is 1. If the form or parameters of F(x) are determined from the data X_i the critical values determined in this way are invalid. In such cases, 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 by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the Gumbel distribution, are also given by Shorak & Wellner (1986, p239).

Two-sample Kolmogorov–Smirnov test

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

and the null hypothesis is rejected at level α if

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).

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_n(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_α such that P(D_n > D_α) = α, then a band of width ±D_α around F_n(x) will entirely contain F(x) with probability 1 − α.

The Kolmogorov–Smirnov statistic in more than one dimension

The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to multivariate data. This is not necessarily straightforward because one may note that the maximum difference between two joint cumulative distribution functions 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 or 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^d−1 such orderings. One such variation is due to Peacock (1983) and another to Fasano & Franceschini (1987): see Lopes et al. (2007) for a comparison and computational details. Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.

See also

• Andrey Kolmogorov
• Non-parametric statistics
• Anderson–Darling test
• Cramér–von Mises test
• Jarque–Bera test
• Kuiper's test
• Lilliefors test
• Mann–Whitney U
• Shapiro–Wilk test
• Siegel–Tukey test
• Donsker theorem

References

1. ^ Stephens, M. A. (1974). "EDF Statistics for Goodness of Fit and Some Comparisons". Journal of the American Statistical Association 69: 730–737. doi:10.2307/2286009. 

References

• Eadie, W.T.; D. Drijard, F.E. James, M. Roos and B. Sadoulet (1971). Statistical Methods in Experimental Physics. Amsterdam: North-Holland. pp. 269–271. 
• Stuart, Alan; Keith Ord and Steven Arnold (1999). Kendall's Advanced Theory of Statistics. 2A. London: Arnold, a member of the Hodder Headline Group. pp. 25.37–25.43. 
• Pearson E.S., Hartley, H.O. (Editors) (1972) Biometrika Tables for Statisticians, Volume II. CUP. ISBN 0-521-06937-8.
• Shorak, G.R., Wellner, J.A. (1986) Empirical Processes with Applications to Statistics, Wiley. ISBN 0-471-86725-X.
• Stephens, M.A. (1979) Test of fit for the logistic distribution based on the empirical distribution function, Biometrika, 66(3), 591-5.
• Peacock, J. A. (1983). "Two-dimensional goodness-of-fit testing in astronomy". Monthly Notices of the Royal Astronomical Society 202: pp. 615–627. [1]
• Fasano, G., Franceschini, A. (1987) A multidimensional version of the Kolmogorov-Smirnov test. Monthly Notices of the Royal Astronomical Society (ISSN 0035-8711), vol. 225, 155–170.[2]
• Lopes, R.H.C., Reid, I., Hobson, P.R. (2007) "The two-dimensional Kolmogorov-Smirnov test". XI International Workshop on Advanced Computing and Analysis Techniques in Physics Research (April 23-27 2007) Amsterdam, the Netherlands. [3]

External links

• Short introduction
• KS test explanation
• JavaScript implementation of one- and two-sided tests
• Online calculator with the K-S test
• Open-source C++ code to compute the Kolmogorov distribution and perform the K-S test

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