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On robust hypothesis testing with respect to Hellinger distance

Eeshan Modak

TL;DR

The paper studies robust hypothesis testing when the true distribution $p$ need not lie in either model $p_1$ or $p_2$, measuring closeness with the Hellinger distance $H^2$. It establishes a minimax slack $\gamma^*$ describing how far $p$ can be from the hypotheses while still enabling a finite-sample decision that chooses the closer model, and proves a lower bound $\gamma^*\ge \frac{\sqrt{2}}{\sqrt{2}-1}$ alongside an existing upper bound $\gamma^*\le \frac{\sqrt{2}+1}{\sqrt{2}-1}$ via Baraud-type tests. The lower bound is tight when $\mathrm{supp}(p_1)$ and $\mathrm{supp}(p_2)$ are disjoint, and the paper extends the approach to a composite testing setting where each hypothesis is a $H^2$-ball around a fixed distribution, using a simple Baraud-style modification. It also analyzes related results for symmetric $\chi^2$ distance and proposes an alternate test for the composite problem based on a Hellinger midpoint, with explicit separation thresholds $r^*=1-\cos(\theta/2)$ (where $\cos\theta = B(p_1,p_2)$). The work highlights open questions for exact characterizations of $\gamma^*$ and potential quantum analogues, and provides concrete tools for robust decision under misspecification in distributional testing.

Abstract

We study the hypothesis testing problem where the observed samples need not come from either of the specified hypotheses (distributions). In such a situation, we would like our test to be robust to this misspecification and output the distribution closer in Hellinger distance. If the underlying distribution is close to being equidistant from the hypotheses, then this would not be possible. Our main result is quantifying how close the underlying distribution has to be to either of the hypotheses. We also study the composite testing problem, where each hypothesis is a Hellinger ball around a fixed distribution. A generalized likelihood ratio test is known to work for this problem. We give an alternate test for the same.

On robust hypothesis testing with respect to Hellinger distance

TL;DR

The paper studies robust hypothesis testing when the true distribution need not lie in either model or , measuring closeness with the Hellinger distance . It establishes a minimax slack describing how far can be from the hypotheses while still enabling a finite-sample decision that chooses the closer model, and proves a lower bound alongside an existing upper bound via Baraud-type tests. The lower bound is tight when and are disjoint, and the paper extends the approach to a composite testing setting where each hypothesis is a -ball around a fixed distribution, using a simple Baraud-style modification. It also analyzes related results for symmetric distance and proposes an alternate test for the composite problem based on a Hellinger midpoint, with explicit separation thresholds (where ). The work highlights open questions for exact characterizations of and potential quantum analogues, and provides concrete tools for robust decision under misspecification in distributional testing.

Abstract

We study the hypothesis testing problem where the observed samples need not come from either of the specified hypotheses (distributions). In such a situation, we would like our test to be robust to this misspecification and output the distribution closer in Hellinger distance. If the underlying distribution is close to being equidistant from the hypotheses, then this would not be possible. Our main result is quantifying how close the underlying distribution has to be to either of the hypotheses. We also study the composite testing problem, where each hypothesis is a Hellinger ball around a fixed distribution. A generalized likelihood ratio test is known to work for this problem. We give an alternate test for the same.
Paper Structure (13 sections, 4 theorems, 46 equations, 2 figures)

This paper contains 13 sections, 4 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Robust Hypothesis Testing
  3. Robustness to imperfect modelling
  4. Our work
  5. Notation and Convention
  6. Problem Setup
  7. Upper Bound on $\gamma^*$
  8. Results: Lower Bound on $\gamma^*$
  9. When is the lower bound tight?
  10. Some results on testing with respect to symmetric $\chi^2$ distance
  11. Alternate Test for the Composite Testing Problem
  12. Discussion
  13. Acknowledgements

Key Result

Theorem 1

For $\gamma \ge \frac{\sqrt{2}+1}{\sqrt{2}-1}$, every class $\mathcal{P}=\{p_1,p_2\}$ is $\gamma$-robustly testable.

Figures (2)

  • Figure 1: $\mathcal{D}_1$ is a family of distributions such that all its members are $\frac{\sqrt{2}}{\sqrt{2}-1}$ times farther to $p_2$ than to $p_1$ in Hellinger distance. Likewise, all the members of $\mathcal{D}_2$ are $\frac{\sqrt{2}}{\sqrt{2}-1}$ times farther to $p_1$ than to $p_2$ in Hellinger distance.
  • Figure 2: An example of $p^{R_1,R_2}$ (perturbed around $p_1$) and $\bar{p}^{R_1,R_2}$ (perturbed around $p_2$) when $R_1=\{2,4\}$ and $R_2=\{1,3\}$.

Theorems & Definitions (8)

  • Theorem 1: Baraud baraud2011estimator
  • proof
  • Theorem 2
  • proof
  • Lemma 1: bousquet2019optimal
  • proof
  • Lemma 2: yu1997assouad
  • proof