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From Asymptotic to Finite-Sample Minimax Robust Hypothesis Testing

Gökhan Gül

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.

From Asymptotic to Finite-Sample Minimax Robust Hypothesis Testing

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.
Paper Structure (33 sections, 16 theorems, 96 equations, 6 figures, 1 algorithm)

This paper contains 33 sections, 16 theorems, 96 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Summary of the paper and its contributions
  4. Outline of the paper
  5. Notations
  6. Linking Asymptotic Minimax Theory to Finite-Sample Tests
  7. Minimax Robustness Concepts
  8. Single-sample minimax robustness (SMR)
  9. Finite-sample minimax robustness (FMR)
  10. Asymptotic minimax robustness (AMR)
  11. Relations Between Finite-Sample and Asymptotic Minimax Robustness
  12. Poblem Formulation
  13. Finite-Sample Minimax Robust Tests via Asymptotic Theory
  14. Total Variation Neighborhood
  15. Band Model
  16. ...and 18 more sections

Key Result

Corollary 2.3

Any of the following conditions is sufficient for eq:thresh-sep to hold: Moreover, $1\Longrightarrow 2 \Longrightarrow 3$, and neither implication is reversible in general.

Figures (6)

  • Figure 1: Least favorable densities (top) and robust likelihood ratio functions (bottom) for (left) total variation distance based uncertainty classes and (right) $\epsilon$-contamination model.
  • Figure 2: Three different pairs of LFDs arising from the band model together with the bounding functions for $\varepsilon\in\{0.2,0.5,1.5\}$.
  • Figure 3: Three different types of Robust LRFs arising from the band model for $\varepsilon\in\{0.2,0.5,1.5,19\}$ together with the nominal LRF.
  • Figure 4: Degenerate Type-B least favorable densities (top) and likelihood ratio functions (bottom) for band model with asymmetric variances.
  • Figure 5: Least favorable densities (top) and the likelihood ratio function (bottom) for moment-constrained uncertainty classes in the given example.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 2.1: Single-sample minimax robustness
  • Corollary 2.3
  • Definition 2.2: Asymptotic minimax robustness
  • Theorem 2.4
  • Theorem 2.5: Equivalence of SMR and $f$-divergence minimization
  • Theorem 2.6: FMR implies AMR
  • Remark 2.1
  • Theorem 2.7: AMR implies FMR conditionally
  • Remark 2.2
  • Lemma 2.8: Uniqueness of $D_u$-maximizer
  • ...and 18 more