KSTests

Kolmogorov-Smirnov tests (KSTests) are nonparametric methods used to assess whether a sample comes from a specified distribution or whether two samples come from the same distribution. They are based on the empirical distribution functions (EDF) of the data and are widely used for goodness-of-fit testing and for comparing distributions without assuming a particular parametric form.

There are two main variants. The one-sample KS test evaluates whether the EDF of a sample matches a specified distribution F0 by computing Dn = sup_x |Fn(x) - F0(x)|, where Fn is the empirical CDF. The two-sample KS test compares two independent samples by computing D = sup_x |F1,n(x) - F2,m(x)|, where F1,n and F2,m are the two EDFs and n, m are their sample sizes. Under the null hypothesis, the distribution of D follows the Kolmogorov distribution in the asymptotic regime, allowing p-values to be obtained from tables or software; exact critical values exist for small samples.

Assumptions and considerations include independent observations and, for the standard critical values, continuous distributions (ties can affect accuracy). If parameters of the reference distribution are estimated from the data, standard KS critical values are not valid, and adjustments (such as the Lilliefors correction) or resampling methods should be used. The KS test is more sensitive to differences near the center of the distribution and may have limited power against certain alternatives, particularly in the tails or with large samples where even small deviations are detected.

Applications cover goodness-of-fit testing and equality of distributions between groups, with implementations commonly available in statistical software (for example, ks.test in R and equivalent functions in Python’s SciPy).