In statistics, the KolmogorovâÂÂSmirnov test (also KâÂÂS test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions. It can be used to test whether a sample came from a given reference probability distribution (one-sample KâÂÂS test), or to test whether or not two samples came from the same distribution (two-sample KâÂÂS test). It is named after Andrey Kolmogorov and Nikolai Smirnov, who developed it in the 1930s.
The KolmogorovâÂÂSmirnov 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.
The two-sample KâÂÂS 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 KolmogorovâÂÂSmirnov 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 ShapiroâÂÂWilk test or AndersonâÂÂDarling test. However, these other tests have their own disadvantages. For instance, the ShapiroâÂÂWilk test is known not to work well in samples with many identical values.
The empirical distribution function F<sub>n</sub> for n independent and identically distributed (i.i.d.) ordered observations X<sub>i</sub> is defined as
where is the indicator function, equal to 1 if and equal to 0 otherwise.
The KolmogorovâÂÂSmirnov statistic for a given cumulative distribution function F(x) is
where sup<sub>x</sub> 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 GlivenkoâÂÂCantelli theorem, if the sample comes from the distribution F(x), then D<sub>n</sub> converges to 0 almost surely in the limit when 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 AndersonâÂÂDarling test statistic) to properly reject the null hypothesis.
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 . Both the form of the KolmogorovâÂÂSmirnov 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 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 when is finite is not very impressive: even when , the corresponding maximum error is about ; this error increases to when and to a totally unacceptable when . However, a very simple expedient of replacing by
in the argument of the Jacobi theta function reduces these errors to , , and respectively; such accuracy would be usually considered more than adequate for all practical applications.
The goodness-of-fit test or the KolmogorovâÂÂSmirnov test can be constructed by using the critical values of the Kolmogorov distribution. This test is asymptotically valid when It rejects the null hypothesis at level if
where K<sub>ñ</sub> is found from
The asymptotic power of this test is 1.
Fast and accurate algorithms to compute the cdf or its complement for arbitrary and , are available from:
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 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<sub>0</sub> (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<sub>0</sub>, 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<sub>0</sub>. 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.
Under the assumption that 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 , it follows that and and hence, the distribution of depends on the null distribution , 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 when is purely discrete or mixed, implemented in C++ and in the KSgeneral package of the R language. The functions <code>disc_ks_test()</code>, <code>mixed_ks_test()</code> and <code>cont_ks_test()</code> 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 <code>PROC NPAR1WAY</code>, Stata <code>ksmirnov</code> implement the KS test under the assumption that is continuous, which is more conservative if the null distribution is actually not continuous (see
).
The KolmogorovâÂÂSmirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. In this case, the KolmogorovâÂÂSmirnov statistic is
where and are the empirical distribution functions of the first and the second sample respectively, and is the supremum function.
For large samples, the null hypothesis is rejected at level if
Where and are the sizes of first and second sample respectively. The value of is given in the table below for the most common levels of
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. ), 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 KolmogorovâÂÂSmirnov 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 KolmogorovâÂÂSmirnov test when comparing two distribution functions.
Two-sample KS tests have been applied in economics to detect asymmetric effects and to study natural experiments.
While the KolmogorovâÂÂSmirnov 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>ñ</sub> such that P(D<sub>n</sub> > D<sub>ñ</sub>) = ñ, then a band of width ñD<sub>ñ</sub> around F<sub>n</sub>(x) will entirely contain F(x) with probability 1 â ñ.
A distribution-free multivariate KolmogorovâÂÂSmirnov 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 KolmogorovâÂÂSmirnov 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 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 KolmogorovâÂÂSmirnov 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<sup>d</sup> â 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.
The KolmogorovâÂÂSmirnov test is implemented in many software programs. Most of these implement both the one and two sampled test.