Testing Against Independence and a R\'enyi Information Measure

TL;DR

This paper characterizes the error-exponent pairs in hypothesis testing between known joint and unknown product distributions, introducing a Renyi information measure and analyzing the optimality of various tests.

Contribution

It introduces a Renyi measure of dependence related to the error-exponent function and analyzes the asymptotic optimality of several hypothesis tests.

Findings

Empirical mutual information, Hoeffding, and generalized likelihood-ratio tests are asymptotically optimal.

A Renyi measure of dependence is shown to be the Fenchel biconjugate of the error-exponent function.

An example demonstrates the error-exponent function can be non-convex.

Abstract

The achievable error-exponent pairs for the type I and type II errors are characterized in a hypothesis testing setup where the observation consists of independent and identically distributed samples from either a known joint probability distribution or an unknown product distribution. The empirical mutual information test, the Hoeffding test, and the generalized likelihood-ratio test are all shown to be asymptotically optimal. An expression based on a Renyi measure of dependence is shown to be the Fenchel biconjugate of the error-exponent function obtained by fixing one error exponent and optimizing the other. An example is provided where the error-exponent function is not convex and thus not equal to its Fenchel biconjugate.