Global power functions of goodness of fit tests

Abstract

usseldorf 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 fromthe corresponding Neym an–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. 1. Introduction. Omnibus tests are commonly used if the specific structure of certain nonparametric alternatives is unknown. Among other justifications, it turns out that they typically are consistent against fixed alternatives and √ n-consistent under sequences of local alternatives of sample size n .F or these reasons, people often trust in goodness of fit tests and these are frequently applied to data of finite sample size. On the other hand, every asymptotic approximation should be understood as an approximation of the underlying finite sample case. Thus the statistician likes to distinguish and to compare the power of different competing tests. The present paper offers a concept for the comparison and justification of different tests by their power functions and level points. It is shown that under certain circumstances every test has a preference for a finite dimensional space of alternatives. Apart fromthis space, the power function is almost flat on balls of alternatives. There exists no test which pays equal attention to an infinite number of orthogonal alternatives. The results do not only hold for asymptotic models but they also hold for concrete alternatives on the real line at finite sample size and their level points uniformly for the sample size. The results are not surprising. Every statistician knows that it is impossible to separate an infinite sequence of different parameters simultaneously if only a finite number of observations is available. The conclusions of the results are two-fold. 1. The statistician should analyze the goodness of fit tests of his computer package in order to get some knowledge and an impression about their preferences.