Nearly Minimax One-Sided Mixture-Based Sequential Tests

TL;DR

This paper develops nearly minimax one-sided mixture-based sequential tests for simple null versus composite alternatives, providing asymptotic optimality results, performance analysis, and simulation validation.

Contribution

It introduces nearly minimax mixture-based stopping rules for sequential testing against composite hypotheses, with theoretical and empirical performance analysis.

Findings

Identified nearly minimax mixture stopping rules for both discrete and continuous alternatives.

Derived high-order asymptotic approximations for stopping time performance.

Validated asymptotic results through simulation experiments.

Abstract

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.