Asymptotically Minimax Robust Likelihood Ratio Test

TL;DR

This paper introduces a comprehensive framework for designing asymptotically minimax robust likelihood ratio tests under various distributional uncertainties, unifying Bayesian and Neyman--Pearson approaches.

Contribution

It provides a systematic derivation of robust tests using minimax theorems, establishes existence and uniqueness, and derives closed-form least favorable distributions.

Findings

Closed-form robust likelihood ratio functions derived

Numerical simulations confirm theoretical properties

Dabak's approach is shown not to produce minimax robust tests

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.