Asymptotic expansions for the power of adaptive rank tests in the one-sample problem

Abstract

In this paper we consider adaptive rank tests for the one-sample problem. Here adaptation means that the score function J of the rank test is estimated from the sample. We restrict attention to cases with a moderate degree of adaptation, in the sense that we require that the estimated J belongs to a one-parameter family J={J_r|rєI⊂R¹}. For the power of adaptive rank tests of this type, we establish asymptotic expansions under contiguous location alternatives, for general estimators S of the parameter r. These expansions are used to compare, in terms of deficiencies, the performance of these adaptive rank tests to that of rank tests with fixed scores. Conditions on S and J_r are given under which the deficiency tends to a finite limit, which is obtained. It is verified that these conditions hold for a particular class of estimators which are related to the sample kurtosis. In this case explicit results are obtained.

This research was supported by the Netherlands Organization for the Advancement of of Pure Research (Z.W.O.).