On the bayes risk incurred by using asymptotic shapes

Abstract

Let X1 X2…denote Independent and Identically distributed random vectors whose common distributions form a multiparameter exponential family, and consider the problem of sequentially testing separated hypotheses. It is known that the sequential procedure which continues sampling until the likelihood ratio statistic for testing one of the hypotheses exceeds a given level approximates the optimal Bayesian procedure, under general conditions on the loss function and prior distribution. Here we ask whether the approximate procedure is Bayes risk efficient--that is, whether the ratio of the Bayes risk of the approximate procedure to the Bayes risk of the optimal procedure approaches one as the cost of samping approaches zero. We show that the answer depends on the choice of certain parameters in the approximation and the dimensions of the hypotheses.