🤖 AI Summary
This paper addresses the optimal decision problem for composite binary hypothesis testing under the Neyman–Pearson framework: maximizing the expected value of a nonlinear function of the detection probability subject to a false-alarm probability constraint. Methodologically, it establishes a novel equivalence—under a generalized Bayesian perspective—between power functions and generalized Bayes rules, enabling the construction of a weighted likelihood ratio test with a single threshold applicable to both composite null and composite alternative hypotheses. The framework unifies treatment of average- and worst-case false-alarm constraints. By leveraging signed measure integration optimization and exponential-family structural analysis, the authors derive an explicit analytical form for the optimal threshold, yielding closed-form solutions within exponential families. Numerical experiments demonstrate substantial improvements in detection performance while rigorously satisfying the prescribed false-alarm constraints.
📝 Abstract
The composite binary hypothesis testing problem within the Neyman-Pearson framework is considered. The goal is to maximize the expectation of a nonlinear function of the detection probability, integrated with respect to a given probability measure, subject to a false-alarm constraint. It is shown that each power function can be realized by a generalized Bayes rule that maximizes an integrated rejection probability with respect to a finite signed measure. For a simple null hypothesis and a composite alternative, optimal single-threshold decision rules based on an appropriately weighted likelihood ratio are derived. The analysis is extended to composite null hypotheses, including both average and worst-case false-alarm constraints, resulting in modified optimal threshold rules. Special cases involving exponential family distributions and numerical examples are provided to illustrate the theoretical results.