On the asymptotic power of the two-sided Kolmogorov-Smirnov test

Abstract

The asymptotic power of the two-sided one-sample Kolmogorov-Smirnov test (KS) is investigated and compared with the power of some other goodness of fit tests (EDF tests). It is found that this test (similar to the ones of Anderson-Darling and Cramér-von Mises) does not behave like a well- balanced procedure for higher-dimensional alternatives: There are only few directions of deviations from the hypothesis for which it is of reasonable asymptotic power. These directions are determined as follows: Employing spectral theory of compact operators a principal components decomposition of the curvature of the asymptotic power function at the hypothesis is established. This decomposition is given in terms of an orthogonal series in the ‘tangent space’ of directions of alternatives. It shows how the respective test distributes its power in the space of all alternatives. The beginning of the series is evaluated numerically for the KS test. Comparison with the curvature of the optimal power function for a given one-dimensional alternative yields local efficiencies of KS which are high for one direction only, and then rapidly decrease to zero. These findings are supported by global upper bounds of the asymptotic power function of EDF tests, which indicate that for directions with small curvature at the origin the whole power function must have bad efficiency properties.