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Asymptotically Minimax Robust Likelihood Ratio Test

Gökhan Gül

TL;DR

This work addresses robust binary hypothesis testing under distributional misspecification in both Bayesian and Neyman–Pearson settings. It introduces a unified framework based on the $u$-affinity $D_u(G_0,G_1)$, proves the existence and uniqueness of least-favorable distributions via Sion's minimax theorem and KKT conditions, and derives closed-parametric LFDs for KL-divergence, $\alpha$-divergence, and symmetric $\alpha$-divergence neighborhoods. It shows that Dabak’s approach fails to achieve asymptotic minimax robustness in the NP sense, while the proposed methodology yields robust likelihood-ratio tests computable from nonlinear systems; simulations validate the theoretical exponents and the rotation-with-clipping behavior of the robust LRFs. The framework provides a systematic, generalizable path to design robust tests under model uncertainty with practical finite-sample implications through tractable nonlinear equations.

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, $α$-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.

Asymptotically Minimax Robust Likelihood Ratio Test

TL;DR

This work addresses robust binary hypothesis testing under distributional misspecification in both Bayesian and Neyman–Pearson settings. It introduces a unified framework based on the -affinity , proves the existence and uniqueness of least-favorable distributions via Sion's minimax theorem and KKT conditions, and derives closed-parametric LFDs for KL-divergence, -divergence, and symmetric -divergence neighborhoods. It shows that Dabak’s approach fails to achieve asymptotic minimax robustness in the NP sense, while the proposed methodology yields robust likelihood-ratio tests computable from nonlinear systems; simulations validate the theoretical exponents and the rotation-with-clipping behavior of the robust LRFs. The framework provides a systematic, generalizable path to design robust tests under model uncertainty with practical finite-sample implications through tractable nonlinear equations.

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, -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.
Paper Structure (27 sections, 8 theorems, 64 equations, 5 figures, 1 table)

This paper contains 27 sections, 8 theorems, 64 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Motivation
  4. Summary of the paper and its contributions
  5. Outline of the paper
  6. Notations
  7. Asymptotic Minimax Robustness
  8. Optimum Threshold
  9. Derivation of Minimax Equations and Problem Formulation
  10. Bayesian Tests
  11. Neyman-Pearson Tests
  12. Saddle Value Condition
  13. Problem Statement
  14. Derivation of Asymptotically Minimax Robust Tests
  15. KL-divergence Neighborhood
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $\boldsymbol{Y}=(Y_1,\ldots,Y_n)$ be a sequence of i.i.d. r.v.s, where each $Y_k$ is distributed as $G_j\in \mathscr{G}_j$ under $\mathcal{H}_j$. Furthermore, let each $X_k$ has a finite moment generating function For the test given by eqgfg, if then, as $n\to\infty$, false alarm and miss detection probabilities decrease exponentially with the rate functions given by

Figures (5)

  • Figure 1: Nominal distributions abbreviated by $d_1$ in Table \ref{['tab1']} and the corresponding LFDs with respect to their densities for $\epsilon_0=\epsilon_1=0.1$, where $u=0.5$ for all $\alpha$.
  • Figure 3: Nominal distributions abbreviated by $d_2$ in Table \ref{['tab1']} and the corresponding LFDs for $\epsilon_0=\epsilon_1=0.1$.
  • Figure 5: Nominal distributions abbreviated by $d_3$ in Table \ref{['tab1']} and the corresponding LFDs with respect to their densities for $\epsilon_0=\epsilon_1=0.05$, where $u=0.46$.
  • Figure 7: $u$-affinity as a function of $u$ for the LFDs obtained for various pairs of distributions as well as uncertainty classes.
  • Figure 9: Asymptotic decay rate of the false alarm probability $I_0$ for the asymptotically minimax robust test ((a)-test) and Dabak's test ((a$^*$)-test).

Theorems & Definitions (12)

  • Theorem 2.1
  • Definition 2.1
  • Remark 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Theorem 4.1
  • Remark 4.1
  • Theorem 4.2
  • Remark 4.2
  • Theorem 4.3
  • ...and 2 more