Frequentist properties of Bayesian sequential tests

Abstract

SUMMARY Let X1, X2, ... denote independent random variables which are normally distributed with unknown mean 0 and unit variance. We consider sequential tests of the hypothesis Ho: 0 6. The tests which we consider were shown by Schwarz (1962) to approximate the optimal Bayesian tests with respect to a general loss structure and any prior density which is everywhere positive. Their continuation regions are bounded subsets of the (n, Sn) plane, where Sn is the cumulative sum. We give both inequalities and asymptotic expressions for the power function and the expected sample size. We also give comparisons of the properties of the approximate Bayesian test, the sequential probability ratio test, and the fixed sample size test.