Nearly Minimax One-Sided Mixture-Based Sequential Tests

We focus on one-sided, mixture-based stopping rules for the problem of sequential testing a simple null hypothesis against a composite alternative. For the latter, we consider two cases—either a discrete alternative or a continuous alternative that can be embedded into an exponential family. For each case, we find a mixture-based stopping rule that is nearly minimax in the sense of minimizing the maximal Kullback–Leibler information. The proof of this result is based on finding an almost Bayes rule for an appropriate sequential decision problem and on high-order asymptotic approximations for the performance characteristics of arbitrary mixture-based stopping times. We also evaluate the asymptotic performance loss of certain intuitive mixture rules and verify the accuracy of our asymptotic approximations with simulation experiments.