Asymptotic Deficiencies of One-Sample Rank Tests Under Restricted Adaptation

In this paper we consider adaptive one-sample rank tests of the following type: the score function $J$ of the test is estimated from the sample under the restriction that $J \in \mathscr{J}$, for some given one-parameter family $\mathscr{J} = \{J_r, r \in I \subset R^1\}$. Using deficiencies, we compare the performance of such tests to that of rank tests with fixed scores. Conditions on the estimator $S$ of the parameter $r$ and on $J_r$ are given, under which the deficiency tends to a finite limit, which is obtained. For a particular class of estimators which are related to the sample kurtosis, explicit results are obtained.