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Text of the page (random words):
displaystyle sqrt n d_ n converges to the kolmogorov distribution which does not depend on f this result may also be known as the kolmogorov theorem the accuracy of this limit as an approximation to the exact cdf of k displaystyle k when n displaystyle n is finite is not very impressive even when n 1000 displaystyle n 1000 the corresponding maximum error is about 0 9 displaystyle 0 9 this error increases to 2 6 displaystyle 2 6 when n 100 displaystyle n 100 and to a totally unacceptable 7 displaystyle 7 when n 10 displaystyle n 10 however a very simple expedient of replacing x displaystyle x by x 1 6 n x 1 4 n displaystyle x frac 1 6 sqrt n frac x 1 4n in the argument of the jacobi theta function reduces these errors to 0 003 displaystyle 0 003 0 027 displaystyle 0 027 and 0 27 displaystyle 0 27 respectively such accuracy would be usually considered more than adequate for all practical applications 6 the goodness of fit test or the kolmogorov smirnov test can be constructed by using the critical values of the kolmogorov distribution this test is asymptotically valid when n displaystyle n to infty it rejects the null hypothesis at level α displaystyle alpha if n d n k α displaystyle sqrt n d_ n k_ alpha where k α is found from pr k k α 1 α displaystyle operatorname pr k leq k_ alpha 1 alpha the asymptotic power of this test is 1 fast and accurate algorithms to compute the cdf pr d n x displaystyle operatorname pr d_ n leq x or its complement for arbitrary n displaystyle n and x displaystyle x are available from 7 and 8 for continuous null distributions with code in c and java to be found in 7 9 for purely discrete mixed or continuous null distribution implemented in the ksgeneral package 10 of the r project for statistical computing which for a given sample also computes the ks test statistic and its p value alternative c implementation is available from 9 test with estimated parameters edit if either the form or the 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 11 and later publications also include the gumbel distribution 12 the lilliefors test represents a special case of this for the normal distribution the logarithm transformation may help to overcome cases where the kolmogorov test data does not seem to fit the assumption that it came from the normal distribution using estimated parameters the question arises which estimation method should be used usually this would be the maximum likelihood method but e g for the normal distribution mle has a large bias error on sigma using a moment fit or ks minimization instead has a large impact on the critical values and also some impact on test power if we need to decide for student t data with df 2 via ks test whether the data could be normal or not then a ml estimate based on h 0 data is normal so using the standard deviation for scale would give much larger ks distance than a fit with minimum ks in this case we should reject h 0 which is often the case with mle because the sample standard deviation might be very large for t 2 data but with ks minimization we may get still a too low ks to reject h 0 in the student t case a modified ks test with ks estimate instead of mle makes the ks test indeed slightly worse however in other cases such a modified ks test leads to slightly better test power 13 discrete and mixed null distribution edit under the assumption that f displaystyle f is non decreasing and right continuous with countable possibly infinite number of jumps the ks test statistic can be expressed as d n sup x f n x f x sup 0 t 1 f n f 1 t f f 1 t displaystyle d_ n sup _ x f_ n x f x sup _ 0 leq t leq 1 f_ n f 1 t f f 1 t from the right continuity of f displaystyle f it follows that f f 1 t t displaystyle f f 1 t geq t and f 1 f x x displaystyle f 1 f x leq x and hence the distribution of d n displaystyle d_ n depends on the null distribution f displaystyle f i e is no longer distribution free as in the continuous case therefore a fast and accurate method has been developed to compute the exact and asymptotic distribution of d n displaystyle d_ n when f displaystyle f is purely discrete or mixed 9 implemented in c and in the ksgeneral package 10 of the r language the functions disc_ks_test mixed_ks_test and cont_ks_test compute also the ks test statistic and p values for purely discrete mixed or continuous null distributions and arbitrary sample sizes the ks test and its p values for discrete null distributions and small sample sizes are also computed in 14 as part of the dgof package of the r language major statistical packages among which sas proc npar1way 15 stata ksmirnov 16 implement the ks test under the assumption that f x displaystyle f x is continuous which is more conservative if the null distribution is actually not continuous see 17 18 19 two sample kolmogorov smirnov test edit 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 ks 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 d n m sup x f 1 n x f 2 m x displaystyle d_ n m sup _ x f_ 1 n x f_ 2 m x where f 1 n displaystyle f_ 1 n and f 2 m displaystyle f_ 2 m are the empirical distribution functions of the first and the second sample respectively and sup displaystyle sup is the supremum function for large samples the null hypothesis is rejected at level α displaystyle alpha if d n m c α n m n m displaystyle d_ n m c alpha sqrt frac n m n cdot m where n displaystyle n and m displaystyle m are the sizes of first and second sample respectively the value of c α displaystyle c alpha is given in the table below for the most common levels of α displaystyle alpha α displaystyle alpha 0 20 0 15 0 10 0 05 0 025 0 01 0 005 0 001 c α displaystyle c alpha 1 073 1 138 1 224 1 358 1 48 1 628 1 731 1 949 and in general 20 by c α ln α 2 1 2 displaystyle c left alpha right sqrt ln left tfrac alpha 2 right cdot tfrac 1 2 so that the condition reads d n m ln α 2 n m 2 n m displaystyle d_ n m sqrt ln left tfrac alpha 2 right cdot frac n m 2n cdot m here again the larger the sample sizes the more sensitive the minimal bound for a given ratio of sample sizes e g m n displaystyle m n the minimal bound scales in the size of either of the samples according to its inverse square root the large sample formula above is only approximate and can be inaccurate for small samples an exact p value can instead be obtained from the permutation conditional distribution of the two sample statistic given the pooled sample which remains valid whether or not the data contain tied observations for continuous data with no ties it coincides with the usual distribution free null distribution this permutation p value can be computed exactly by a recurrence for arbitrary sample sizes 21 22 23 for continuous distributions the permutation statistic and the ordinary statistic d n m displaystyle d_ n m share the same limiting null distribution so their critical values agree asymptotically 24 25 this asymptotic agreement has been shown to extend to arbitrary underlying distributions including discrete or mixed data with ties and to weighted versions of the statistic whose weight function can be chosen to improve power 26 23 these methods are implemented in the ks2sample function of the ksgeneral package 23 10 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 whether it s normal or not normal again tables of critical values have been published a shortcoming of the univariate kolmogorov smirnov test is that it is not very powerful because it is devised to be sensitive against all possible types of differences between two distribution functions some argue 27 28 that the cucconi test originally proposed for simultaneously comparing location and scale can be much more powerful than the kolmogorov smirnov test when comparing two distribution functions setting confidence limits for the shape of a distribution function edit main article dvoretzky kiefer wolfowitz inequality 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 edit a distribution free multivariate kolmogorov smirnov goodness of fit test has been proposed by justel peña and zamar 1997 29 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 data this is not straightforward because 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 pr x x y y displaystyle pr x x land y y or pr x x y y displaystyle pr x x land y y 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 ks statistics in d dimensions there are 2 d 1 such orderings one such variation is due to peacock 30 see also gosset 31 for a 3d version and another to fasano and franceschini 32 see lopes et al for a comparison and computational details 33 critical values for the test statistic can be obtained by simulations but depend on the dependence structure in the joint distribution implementations edit the kolmogorov smirnov test is implemented in many software programs most of these implement both the one and two sampled test mathematica has kolmogorovsmirnovtest matlab s statistics toolbox has kstest and kstest2 for one sample and two sample kolmogorov smirnov tests respectively the r package ksgeneral 10 computes the ks test statistics and its p values under arbitrary possibly discrete mixed or continuous null distribution r s statistics base package implements the test as ks test stats in its stats package sas implements the test in its proc npar1way procedure in python the scipy package implements the test in the scipy stats kstest function 34 systat spss inc chicago il java has an implementation of this test provided by apache commons 35 knime has a node implementing this test based on the above java implementation 36 julia has the package hypothesistests jl with the function exactonesamplekstest x abstractvector real d univariatedistribution 37 statsdirect statsdirect ltd manchester uk implements all common variants stata stata corporation college station tx implements the test in ksmirnov kolmogorov smirnov equality of distributions test command 38 pspp implements the test in its kolmogorov smirnov or using ks shortcut function the real statistics resource pack for excel runs the test as kscrit and ksprob 39 clickhouse implements the test in its kolmogorovsmirnovtest function see also edit lepage test cucconi test kuiper s test shapiro wilk test anderson darling test cramér von mises test wasserstein metric references edit 7 2 1 2 kolmogorov smirnov test retrieved 8 october 2025 stephens m a 1974 edf statistics for goodness of fit and some comparisons journal of the american statistical association 69 347 730 737 doi 10 2307 2286009 jstor 2286009 marsaglia g tsang ww wang j 2003 evaluating kolmogorov s distribution journal of statistical software 8 18 1 4 doi 10 18637 jss v008 i18 1 2 kolmogorov a 1933 sulla determinazione empirica di una legge di distribuzione g ist ital attuari 4 83 91 smirnov n 1948 table for estimating the goodness of fit of empirical distributions annals of mathematical statistics 19 2 279 281 doi 10 1214 aoms 1177730256 vrbik jan 2018 small sample corrections to kolmogorov smirnov test statistic pioneer journal of theoretical and applied statistics 15 1 2 15 23 1 2 simard r l ecuyer p 2011 computing the two sided kolmogorov smirnov distribution journal of statistical software 39 11 1 18 doi 10 18637 jss v039 i11 moscovich a nadler b 2017 fast calculation of boundary crossing probabilities for poisson processes statistics and probability letters 123 177 182 arxiv 1503 04363 doi 10 1016 j spl 2016 11 027 s2cid 12868694 1 2 3 dimitrova ds kaishev vk tan s 2020 computing the kolmogorov smirnov distribution when the underlying cdf is purely discrete mixed or continuous journal of statistical software 95 10 1 42 doi 10 18637 jss v095 i10 1 2 3 4 dimitrova dimitrina yun jia kaishev vladimir tan senren 21 may 2024 ksgeneral ksgeneral computing p values of the one sample k s test and the two sample k s and kuiper tests for dis continuous null distribution cran r project org package ksgeneral pearson e s hartley h o eds 1972 biometrika tables for statisticians vol 2 cambridge university press pp 117 123 tables 54 55 isbn 978 0 521 06937 3 shorack galen r wellner jon a 1986 empirical processes with applications to statistics wiley p 239 isbn 978 0 471 86725 8 a modified kolmogorov smirnov test for normality pdf munich personal repec archive retrieved 10 september 2026 arnold taylor b emerson john w 2011 nonparametric goodness of fit tests for discrete null distributions pdf the r journal 3 2 34 dash 39 doi 10 32614 rj 2011 016 sas stat r 14 1 user s guide support sas com retrieved 14 april 2018 ksmirnov kolmogorov smirnov equality of distributions test pdf stata com retrieved 14 april 2018 noether ge 1963 note on the kolmogorov statistic in the discrete case metrika 7 1 115 116 doi 10 1007 bf02613966 s2cid 120687545 slakter mj 1965 a comparison of the pearson chi square and kolmogorov goodness of fit tests with respect to validity journal of the american statistical association 60 311 854 858 doi 10 2307 2283251 jstor 2283251 walsh je 1963 bounded probability properties of kolmogorov smirnov and similar statistics for discrete data annals of the institute of statistical mathematics 15 1 153 158 do...