Finite Sample Size Optimality of GLR Tests

TL;DR

This paper introduces a joint framework for hypothesis testing and parameter estimation, demonstrating that the GLR test is optimal for finite samples under this combined criterion.

Contribution

It proposes a unified approach combining Neyman-Pearson and Bayesian criteria for joint detection and estimation, revealing the optimality of the GLR test in finite samples.

Findings

GLR test is finite-sample-size optimal under the combined criterion

New detection/estimation structures emerge from the joint approach

Framework unifies detection and estimation for improved performance

Abstract

In several interesting applications one is faced with the problem of simultaneous binary hypothesis testing and parameter estimation. Although such joint problems are not infrequent, there exist no systematic analysis in the literature that treats them effectively. Existing approaches consider the detection and the estimation subproblems separately, applying in each case the corresponding optimum strategy. As it turns out the overall scheme is not necessarily optimum since the criteria used for the two parts are usually incompatible. In this article we propose a mathematical setup that considers the two problems jointly. Specifically we propose a meaningful combination of the Neyman-Pearson and the Bayesian criterion and we provide the optimum solution for the joint problem. In the resulting optimum scheme the two parts interact with each other, producing detection/estimation structures that are completely novel. Notable side-product of our work is the proof that the well known GLR test is finite-sample-size optimum under this combined sense.