Finite-sample expansions for the optimal error probability in asymmetric binary hypothesis testing
Valentinian Lungu, Ioannis Kontoyiannis
TL;DR
This work derives sharp finite-n bounds for the optimal first-error probability in asymmetric binary hypothesis testing between $P$ and $Q$ with i.i.d. observations. By fusing large deviations with Gaussian (Berry–Esseen) type approximations and employing a tilted family $Q_{\alpha}$ to express the error-exponent function $D(\delta)$, it establishes concrete nonasymptotic bounds: $\log E_{1}^*(n,\delta) = -n D(\delta) - \frac{1}{2(1-\alpha^*)}\log n + O(1)$, with explicit constants depending on moments under $Q_{\alpha^*}$. In the asymmetric regime, these finite-n expansions outperform traditional normal-approximation and pure exponent bounds, providing accurate, computable estimates for moderate $n$ and small $\epsilon$. The results unify several classical findings under a single finite-sample framework and have practical implications for sample complexity and decision rules in binary testing problems.
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.