Kolmogorov–Smirnov test
In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be 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). In essence, the test answers the question "What is the probability that this collection of samples could have been drawn from that probability distribution?" or, in the second case, "What is the probability that these two sets of samples  were drawn from the same (but unknown) probability distribution?". It is named after Andrey Kolmogorov and Nikolai Smirnov. 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 sample is drawn from the reference distribution (in the one-sample case) or that the samples are drawn from the same distribution (in the two-sample case). In the one-sample case, the distribution considered  under the null hypothesis may be continuous (see Section 2), purely discrete or mixed (see Section 2.2). In the two-sample case (see Section 3), the distribution considered under the null hypothesis is a continuous distribution but is otherwise unrestricted. However, the two sample test can also be performed under more general conditions that allow for discontinuity, heterogeneity and dependence across samples. The two-sample K–S 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 these to define the specific reference distribution changes the null distribution of the test statistic (see Test with estimated parameters). 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. However, these other tests have their own disadvantages. For instance the Shapiro–Wilk test is known not to work well in samples with many identical values. Kolmogorov–Smirnov statistic. The empirical distribution function "F" for "n" independent and identically distributed (i.i.d.) ordered observations "X" is defined as The Kolmogorov–Smirnov statistic for a given cumulative distribution function "F"("x") is where sup is the supremum of the set of distances. Intuitively, the statistic takes the largest absolute difference between the two distribution functions across all "x" values. By the Glivenko–Cantelli  theorem, if the sample comes from distribution "F"("x"), then "D" converges to 0 almost surely in the limit when formula_5 goes to infinity. Kolmogorov strengthened this result, by effectively providing the rate of this convergence (see Kolmogorov distribution). Donsker's theorem provides a yet stronger result. In practice, the statistic requires a relatively large number of data points (in comparison to other goodness of fit criteria such as the Anderson–Darling test statistic) to properly reject the null hypothesis. 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 which can also be expressed by the Jacobi theta function formula_8. Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Smirnov. Recurrence relations for the distribution of the test statistic in finite samples are available. 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 formula_10 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 formula_11 when formula_5 is finite is not very impressive: even when formula_13, the corresponding maximum error is about formula_14; this error increases to formula_15 when formula_16 and to a totally unacceptable formula_17 when formula_18. However, a very simple expedient of replacing formula_19 by in the argument of the Jacobi theta function reduces these errors to formula_21, formula_22, and formula_23 respectively; such accuracy would be usually considered more than adequate for all practical  applications. 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 formula_24. It rejects the null hypothesis at level formula_25 if where "K" is found from The asymptotic power of this test is 1. Fast and accurate algorithms to compute the cdf formula_28 or its complement for arbitrary formula_5 and formula_19, are available from: Test with estimated parameters. If either the form or the parameters of "F"("x") are determined from the data "X" 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, and later publications also include the Gumbel distribution. 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 (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, 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. 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. Discrete and mixed null distribution. Under the assumption that formula_31 is non-decreasing and right-continuous, with countable (possibly infinite ) number of jumps, the KS test statistic can be expressed as: From the right-continuity of formula_31, it follows that formula_34 and formula_35 and hence, the distribution of formula_36 depends on the null distribution formula_31, 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 formula_36 when formula_31 is purely discrete or mixed, implemented in C++ and in the KSgeneral package of the R language. The functions codice_1, codice_2 and codice_3 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 as part of the dgof package of the R language. Major statistical packages among which SAS codice_4, Stata codice_5 implement the KS test under the assumption that formula_31 is continuous, which is more conservative if the null distribution is actually not continuous (see ). 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 where formula_42 and formula_43 are the empirical distribution functions of the first and the second sample respectively, and formula_44 is the supremum function. For large samples, the null hypothesis is rejected at level formula_25 if Where formula_5 and formula_48 are the sizes of first and second sample respectively. The value of formula_49 is given in the table below for the most common levels of formula_25 and in general by so that the condition reads Here, again, the larger the sample sizes, the more sensitive the minimal bound: For a given ratio of sample sizes (e.g. formula_53), the  minimal bound scales in the size of either of the samples according to its inverse square root. 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 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. While the Kolmogorov–Smirnov test is usually used to test whether a given "F"("x") is the underlying probability distribution of "F"("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" such that P("D" > "D") = "α", then a band of width ±"D" around "F"("x") will entirely contain "F"("x") with probability 1 − "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). 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 formula_54 or formula_55 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 2−1 such orderings. One such variation is due to Peacock (see also Gosset for a 3D version) and another to Fasano and Franceschini (see Lopes et al. 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. In one dimension, the Kolmogorov–Smirnov statistic is identical to the so-called star discrepancy D, so another native KS extension to higher dimensions would be simply to use D also for higher dimensions. Unfortunately, the star discrepancy is hard to calculate in high dimensions . In 2021 the functional form of the multivariate KS test statistic was proposed, which simplified the problem of estimating the tail probabilities of the multivariate KS test statistic, which is needed for the statistical test. For the multivariate case, if "F" is the "i"th continuous marginal from a probability distribution with "k" variables, then so the limiting distribution does not depend on the marginal distributions. Implementations. The Kolmogorov-Smirnov test is implemented in many software programs. Most of these implement both the one and two sampled test.