🤖 AI Summary
This work addresses the finite-sample minimax robust hypothesis testing problem under distributional uncertainty by rigorously bridging finite-sample and asymptotic optimal solutions through asymptotic theory, thereby avoiding conventional heuristic constructions. The authors model uncertainty using total variation distance and band models, and derive explicit parametric forms for the least favorable distributions and the robust likelihood ratio function. Theoretical analysis establishes that, under both uncertainty models, the finite-sample minimax robust test coincides with its asymptotic counterpart, extending existing results to asymmetric robust parameter settings and providing a systematic unification of prior approaches. Numerical simulations corroborate the theoretical guarantees and demonstrate the practical efficacy of the proposed test.
📝 Abstract
This paper establishes a formal connection between finite-sample and asymptotically minimax robust hypothesis testing under distributional uncertainty. It is shown that, whenever a finite-sample minimax robust test exists, it coincides with the solution of the corresponding asymptotic minimax problem. This result enables the analytical derivation of finite-sample minimax robust tests using asymptotic theory, bypassing the need for heuristic constructions. The total variation distance and band model are examined as representative uncertainty classes. For each, the least favorable distributions and corresponding robust likelihood ratio functions are derived in parametric form. In the total variation case, the new derivation generalizes earlier results by allowing unequal robustness parameters. The theory also explains and systematizes previously heuristic designs. Simulations are provided to illustrate the theoretical results.