From Asymptotic to Finite-Sample Minimax Robust Hypothesis Testing

TL;DR

This paper links finite-sample and asymptotic minimax robust hypothesis testing, enabling analytical derivation of finite-sample tests from asymptotic theory under distributional uncertainty.

Contribution

It establishes that finite-sample minimax tests coincide with asymptotic solutions when they exist, simplifying their derivation and understanding.

Findings

Finite-sample minimax tests match asymptotic solutions when available.

Derived least favorable distributions for total variation and band models.

Generalized total variation results to unequal robustness parameters.

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.