Asymptotic Solutions of the Sequential Compound Decision Problem

This paper considers the sequential compound decision problem for the case where the component decisions are of the simple versus simple hypothesis testing type, and thus can be stated in terms of testing whether = 0 or = 1. From this point of view it may be considered to be the sequential analogue of 3. The risk for the compound decision is taken to be the average of the risks in the component problems. We prove that whenever the Bayes envelope power function of the component has a derivative, the decision function which at each stage plays Bayes against an estimate of the average of the previous parameter values, has a risk which for all n sufficiently large lies uniformly close to the envelope risk function. We also exhibit a randomized version of the above decision function which has this property for all possible envelope functions.