Monotonicity of Bayes Sequential Tests

Abstract

Consider the problem of sequential testing of a one sided hypothesis when the risk function is a linear combination of a probability of an error component and an expected cost component. Sobel's results on monotonicity of Bayes procedures and essentially complete classes are extended. Sufficient conditions are given for every Bayes test to be monotone. The conditions are satisfied when the observations are from an exponential family. They are also satisfied for orthogonally invariant tests of a mean vector of a multivariate normal distribution and for scale invariant tests of two normal variances. Essentially complete classes of tests are the monotone tests for all situations where these sufficient conditions are satisfied.