Testing composite hypotheses via convex duality

TL;DR

This paper introduces a convex duality framework for testing composite hypotheses, providing necessary and sufficient optimality conditions that extend classical results and differ from previous duality approaches.

Contribution

It develops a novel convex duality method using Fenchel duality to derive optimality conditions for composite hypothesis testing, advancing theoretical understanding.

Findings

Provides necessary and sufficient conditions for optimal tests

Extends classical hypothesis testing results with convex duality

Offers a new approach differing from prior duality methods

Abstract

We study the problem of testing composite hypotheses versus composite alternatives, using a convex duality approach. In contrast to classical results obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289--302), where sufficient optimality conditions are derived via Lagrange duality, we obtain necessary and sufficient optimality conditions via Fenchel duality under compactness assumptions. This approach also differs from the methodology developed in Cvitani\'{c} and Karatzas (Bernoulli 7 (2001) 79--97).