Asymptotically Optimal Tests for One- and Two-Sample Problems

TL;DR

This paper demonstrates the asymptotic optimality of Hoeffding's likelihood ratio test for one- and two-sample hypothesis testing, providing new proofs and extending the results to the two-sample case.

Contribution

The paper offers a streamlined proof of Hoeffding's test optimality and extends it to two-sample testing with a strong converse result.

Findings

Hoeffding's likelihood ratio test is asymptotically optimal for one-sample testing.

A similar optimal test is derived for two-sample testing based on empirical relative entropy.

A strong converse for the two-sample test is established.

Abstract

In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.