Asymptotically Minimax Robust Hypothesis Testing
This work addresses robustness in statistical hypothesis testing, generalizing earlier theories to allow analytical derivations and resolve unanswered questions, though it appears incremental in extending existing frameworks.
The paper formalizes asymptotically minimax robust hypothesis testing for Bayesian and Neyman-Pearson tests, proving existence and uniqueness of such tests using minimax theorems and deriving least favorable distributions in parametric forms.
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, $α$-divergence, symmetrized $α$-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.