The Bahadur Efficiency of Tests of Some Joint Hypotheses

This article examines the relative efficiency of finite induced and Wald-like (or "infinite induced") tests of some commonly encountered joint hypotheses. One two-sided testing problem and two types of two-sided testing problems concerning two parameters and normally distributed estimates are considered. The exact slopes of six test statistics are derived, and from these the Bahadur relative efficiency of tests based on any pair of the statistics can be easily computed. For the problems considered, Bahadur relative efficiency is equal to limiting Pitman efficiency, whereas Pitman relative efficiency coincides with exact finite sample relative efficiency. I present numerical results showing that Bahadur relative efficiency generally approximates Pitman efficiency rather poorly, although the patterns exhibited by these two efficiency criteria bear a reasonable similarity in most cases.