Finite-sample expansions for the optimal error probability in asymmetric binary hypothesis testing

TL;DR

This paper derives sharp finite-sample bounds and accurate nonasymptotic expansions for the minimal error probability in asymmetric binary hypothesis testing, improving upon previous approximation methods.

Contribution

It introduces new bounds and expansions that are more precise for asymmetric hypothesis testing, utilizing large deviations and Gaussian approximation techniques.

Findings

New bounds outperform existing approximations in accuracy.

Explicit constants are provided for nonasymptotic error probability estimates.

Examples demonstrate significant improvements over normal approximation and error exponent methods.

Abstract

The problem of binary hypothesis testing between two probability measures is considered. New sharp bounds are derived for the best achievable error probability of such tests based on independent and identically distributed observations. Specifically, the asymmetric version of the problem is examined, where different requirements are placed on the two error probabilities. Accurate nonasymptotic expansions with explicit constants are obtained for the error probability, using tools from large deviations and Gaussian approximation. Examples are shown indicating that, in the asymmetric regime, the approximations suggested by the new bounds are significantly more accurate than the approximations provided by either of the two main earlier approaches -- normal approximation and error exponents.