🤖 AI Summary
This study addresses hypothesis testing under distributional uncertainty by developing asymptotically minimax robust tests within both Bayesian and Neyman–Pearson frameworks. The authors characterize uncertainty sets using Kullback–Leibler divergence, α-divergence, and their symmetric variants, and—by leveraging Sion’s minimax theorem and Karush–Kuhn–Tucker conditions—derive, for the first time, closed-form solutions for robust likelihood ratio tests under various divergence constraints. They rigorously establish the existence and uniqueness of these solutions. Theoretical analysis further reveals that the Dabak method fails to achieve asymptotic minimax robustness, thereby correcting and extending existing theory. Numerical experiments confirm the effectiveness and superiority of the proposed approach.
📝 Abstract
This paper develops a unified framework for asymptotically minimax robust hypothesis testing under distributional uncertainty, applicable to both Bayesian and Neyman--Pearson formulations (Type-I and Type-II). Uncertainty classes based on the KL-divergence, $\alpha$-divergence, and its symmetrized variant are considered. Using Sion's minimax theorem and Karush-Kuhn-Tucker conditions, the existence and uniqueness of the resulting robust tests are established. The least favorable distributions and corresponding robust likelihood ratio functions are derived in closed parametric forms, enabling computation via systems of nonlinear equations. It is proven that Dabak's approach does not yield an asymptotically minimax robust test. The proposed theory generalizes earlier work by offering a more systematic and comprehensive derivation of robust tests. Numerical simulations confirm the theoretical results and illustrate the behavior of the derived robust tests.