Bahadur Optimality of Sequential Experiments for Exponential Families

Abstract

A theorem of Bahadur on the asymptotic optimality of the likelihood ratio statistic has been extended to sequential analysis by Berk and Brown (1978) in the context of testing one-sided hypotheses about the mean of a normal distribution with known variance. In this work, Bahadur's theorem is extended to sequential analysis for general hypotheses about the parameters of an exponential family of distributions. Specifically, it is shown that, under certain conditions, modifications of the likelihood ratio statistic analogous to those exhibited by Berk and Brown (1978) in the above normal context are optimal for any family of stopping times approaching $\infty$. These results indicate that Bahadur efficiency has a limited impact in sequential analysis.