Asymptotically Minimax Robust Hypothesis Testing

TL;DR

This paper formalizes the design of asymptotically minimax robust hypothesis tests across various divergence-based uncertainty classes, providing theoretical proofs, explicit formulas, and simulations.

Contribution

It introduces a comprehensive theoretical framework for asymptotically minimax robust hypothesis testing, including existence, uniqueness, and explicit forms of least favorable distributions.

Findings

Proves existence and uniqueness of robust tests using minimax theorems

Derives parametric forms of least favorable distributions and likelihood ratios

Shows Dabak's design is not asymptotically minimax robust

Abstract

The design of asymptotically minimax robust hypothesis testing is formalized for the Bayesian and Neyman-Pearson tests of Type-I and Type-II. The uncertainty classes based on the KL-divergence, $\alpha$-divergence, symmetrized $\alpha$-divergence, total variation distance, as well as the band model, moment classes and p-point classes are considered. Implications between single-sample-, all-sample- and asymptotic minimax robustness are derived. Existence and uniqueness of asymptotically minimax robust tests are proven using Sion's minimax theorem and the Karush-Kuhn-Tucker multipliers. The least favorable distributions and the corresponding robust likelihood ratio functions are derived in parametric forms, which can then be determined by solving a system of equations. The proposed theory proves that Dabak's design does not produce any asymptotically minimax robust test. Furthermore, it also generalizes the earlier works by Huber and Kassam by allowing analytical derivations, hence, providing answers to the questions 'how?', which were left unanswered. Simulations are provided to exemplify and evaluate the theoretical derivations.