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

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={Jr|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 Jr 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.