On minimizing the risk in certain sequential tests

Abstract

For tests of two values of a parameter θ from a family of distributions indexed by θ, the class of sequential probability ratio tests is considered, including procedures in which no observations are taken. With a simple loss structure, the Bayes risk for given error probabilities is minimized in this class; in this paper, the nominal risk is minimized without constraints on the error probabilities, and the results compared numerically with approximations of Chernoff. This is done for θ in a general class of distributions, in more detail when θ is the mean of a multinormal distribution and losses for wrong decision are equal, and also for tests in which the error probabilities are equal.