A Kolmogorov-Smirnov test for \(r\) samples. (English) Zbl 1241.62063

Böhm, Walter; Hornik, Kurt

Fundam. Inform. 117, No. 1-4, 103-125 (2012).

Summary: We consider the problem of testing whether \(r\geq 2\) samples are drawn from the same continuous distribution \(F(x)\). The test statistic we study in some detail is defined as the maximum of the circular differences of the empirical distribution functions, a generalization of the classical 2-sample Kolmogorov-Smirnov test to \(r\geq 2\) independent samples. For the case of equal sample sizes we derive the exact null distribution by counting lattice paths confined to stay in the scaled alcove \(A_r\) of the affine Weyl group \(A_{r - 1}\). This is done using a generalization of the classical reflection principle. By a standard diffusion scaling we derive also the asymptotic distribution of the test statistic in terms of a multivariate Dirichlet series. When the sample sizes are not equal the reflection principle no longer works, but we are able to establish a weak convergence result even in this case showing that by a proper rescaling a test statistic based on a linear transformation of the circular differences of the empirical distribution functions has the same asymptotic distribution as the test statistic in the case of equal sample sizes.

MSC:

62G10	Nonparametric hypothesis testing
62E20	Asymptotic distribution theory in statistics
62G30	Order statistics; empirical distribution functions
62E15	Exact distribution theory in statistics
20F55	Reflection and Coxeter groups (group-theoretic aspects)
62Q05	Statistical tables

Keywords:

lattice path counting; reflection principle; affine Weyl groups; asymptotics distrubutions