KStest

KStest refers to the Kolmogorov–Smirnov test, a nonparametric method for evaluating whether a sample comes from a reference distribution or whether two samples come from the same distribution. It is widely used for goodness-of-fit testing and nonparametric two-sample comparisons. The test centers on the maximum difference between empirical distribution functions.

In a one-sample KS test, the null hypothesis is that the observed data follow a specified continuous distribution F. In a two-sample KS test, the null hypothesis is that the two samples are drawn from the same distribution. The test statistic D is defined as the maximum absolute difference between the empirical distribution functions of the samples (or between the empirical and reference CDF in the one-sample case).

P-values are calculated from the limiting distribution of D under the null hypothesis. For large samples, asymptotic theory provides p-values; for small samples, exact p-values may be available or approximated via simulation. The test can be affected by the presence of ties or estimated parameters in the reference distribution, in which case simulation-based or adjusted methods are often preferred.

KS tests are nonparametric and distribution-free under the null, but have sensitivity characteristics: they are most powerful against differences in location or shape around the center of the distribution and may be less sensitive in the tails. They are commonly implemented in statistical software under various function names, including R's ks.test, Python's scipy.stats.kstest and scipy.stats.ks_2samp, and MATLAB's kstest and kstest2.