Global power functions of goodness of fit tests

Abstract

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