Second-Order Asymptotics of Hoeffding-Like Hypothesis Tests

TL;DR

This paper analyzes the second-order asymptotics of Hoeffding-like hypothesis tests, extending the classical Hoeffding test by replacing the divergence measure and characterizing error rates beyond the first order.

Contribution

It provides a comprehensive characterization of second-order error terms for a broad class of divergence-based tests, including the classical Hoeffding test.

Findings

Derived second-order error asymptotics for Hoeffding-like tests

Extended analysis to a large class of divergence measures

Unified framework encompassing classical Hoeffding test results

Abstract

We consider a binary statistical hypothesis testing problem, where $n$ independent and identically distributed random variables $Z^n$ are either distributed according to the null hypothesis $P$ or the alternate hypothesis $Q$, and only $P$ is known. For this problem, a well-known test is the Hoeffding test, which accepts $P$ if the Kullback-Leibler (KL) divergence between the empirical distribution of $Z^n$ and $P$ is below some threshold. In this paper, we consider Hoeffding-like tests, where the KL divergence is replaced by other divergences, and characterize, for a large class of divergences, the first and second-order terms of the type-II error for a fixed type-I error. Since the considered class includes the KL divergence, we obtain the second-order term of the Hoeffiding test as a special case.