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Optimal Decision Rules for Composite Binary Hypothesis Testing under Neyman-Pearson Framework

Yanglei Song, Berkan Dulek, Sinan Gezici

TL;DR

This work analyzes non-asymptotic composite binary hypothesis testing under the Neyman-Pearson framework with nonlinear objectives on detection probability. It shows that every achievable power function can be realized by a generalized Bayes rule, enabling single-threshold or threshold-with-randomization decision rules based on integrated likelihood ratios. The authors derive integrated and supremum false-alarm controls for composite nulls and alternatives, including clean exponential-family specializations and numerical demonstrations for Normal and Binomial models. The results hinge on reducing optimal rules to a threshold on a function H(y) of the data, with least-favorable-distribution arguments underpinning worst-case analyses. The framework supports nonlinear utility-inspired objectives and robust performance criteria, with practical implications for behavioral-utility detection and robust hypothesis testing.

Abstract

The composite binary hypothesis testing problem within the Neyman-Pearson framework is considered. The goal is to maximize the expectation of a nonlinear function of the detection probability, integrated with respect to a given probability measure, subject to a false-alarm constraint. It is shown that each power function can be realized by a generalized Bayes rule that maximizes an integrated rejection probability with respect to a finite signed measure. For a simple null hypothesis and a composite alternative, optimal single-threshold decision rules based on an appropriately weighted likelihood ratio are derived. The analysis is extended to composite null hypotheses, including both average and worst-case false-alarm constraints, resulting in modified optimal threshold rules. Special cases involving exponential family distributions and numerical examples are provided to illustrate the theoretical results.

Optimal Decision Rules for Composite Binary Hypothesis Testing under Neyman-Pearson Framework

TL;DR

This work analyzes non-asymptotic composite binary hypothesis testing under the Neyman-Pearson framework with nonlinear objectives on detection probability. It shows that every achievable power function can be realized by a generalized Bayes rule, enabling single-threshold or threshold-with-randomization decision rules based on integrated likelihood ratios. The authors derive integrated and supremum false-alarm controls for composite nulls and alternatives, including clean exponential-family specializations and numerical demonstrations for Normal and Binomial models. The results hinge on reducing optimal rules to a threshold on a function H(y) of the data, with least-favorable-distribution arguments underpinning worst-case analyses. The framework supports nonlinear utility-inspired objectives and robust performance criteria, with practical implications for behavioral-utility detection and robust hypothesis testing.

Abstract

The composite binary hypothesis testing problem within the Neyman-Pearson framework is considered. The goal is to maximize the expectation of a nonlinear function of the detection probability, integrated with respect to a given probability measure, subject to a false-alarm constraint. It is shown that each power function can be realized by a generalized Bayes rule that maximizes an integrated rejection probability with respect to a finite signed measure. For a simple null hypothesis and a composite alternative, optimal single-threshold decision rules based on an appropriately weighted likelihood ratio are derived. The analysis is extended to composite null hypotheses, including both average and worst-case false-alarm constraints, resulting in modified optimal threshold rules. Special cases involving exponential family distributions and numerical examples are provided to illustrate the theoretical results.

Paper Structure

This paper contains 21 sections, 15 theorems, 87 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Structural Results
  4. Generalized Bayes Rules
  5. General Structural Results
  6. Simple Null versus Composite Alternative
  7. Exponential Family Distributions
  8. Numerical Example - Normal Distributions
  9. Numerical Example - Binomial Distributions
  10. Discussions
  11. Composite Null and Alternative: Integrated False-Alarm Control
  12. Numerical Example
  13. Composite Null and Alternative: supremum False-Alarm Control
  14. Numerical Example
  15. Conclusion
  16. ...and 6 more sections

Key Result

Lemma 1

Suppose Assumption assumption:continuity holds.

Figures (2)

  • Figure 1: The value of the objective in \ref{['def:object_ell']} as a function $\ell$ for $\beta =2/3$ and $v = 0.69$.
  • Figure 2: The value of the objective in \ref{['def:binomial_prospect']} as a function $p_{\ell}$ for $\ell = 2$, $\beta =1.05$, and $v = 0.69$.

Theorems & Definitions (51)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Theorem 3
  • proof
  • Remark 4
  • Corollary 4
  • ...and 41 more