Pitman Efficiencies of Sequential Tests and Uniform Limit Theorems in Nonparametric Statistics

Abstract

In this paper Pitman's method of constructing and comparing tests based on statistics which are asymptotically normal under the null hypothesis and the local alternatives is extended to sequential tests of statistical hypotheses. The asymptotic normality assumption in Pitman's theory is replaced in its sequential analogue by the weak convergence of normalized processes formed from these statistics under the null hypothesis and the local alternatives. Uniform invariance principles are developed for a large class of statistics, and as an immediate corollary of these results, the desired weak convergence assumption is shown to hold. Furthermore uniform large deviation theorems are obtained for the test statistics and these results guarantee that the sequential tests under consideration have finite expected sample sizes under the null hypothesis and the local alternatives. As an illustration of the general method, the two-sample location problem is studied in detail, and the asymptotic relative efficiencies of the sequential Wilcoxon test, the sequential van der Waerden test and the sequential normal scores test relative to the two-sample sequential $t$-test are easily obtained since one of our key results (Theorem 1) implies that the asymptotic relative efficiencies of these sequential tests coincide with the corresponding Pitman efficiencies of their nonsequential analogues.