Global power functions of goodness of fit tests

Abstract

It is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace. The present benchmark is here the upper one-sided (or two-sided) envelope power function. Every choice of a test fixes a priori a finite dimensional region with high power. It turns out that also the level points are far away from the corresponding Neyman–Pearson test level points except for a finite number of orthogonal directions of alternatives. For certain submodels the result is independent of the underlying sample size. In the last section the statistical consequences and special goodness of fit tests are discussed.