Asymptotically Optimal Tests for One- and Two-Sample Problems

2026-01-16

AI Summary

This work addresses one-sample and two-sample hypothesis testing under unknown distributions by constructing binary tests based on relative entropy thresholds. Leveraging empirical distributions, large deviation theory, and information-theoretic tools, the authors develop a streamlined proof framework that intuitively reveals the asymptotic optimality of Hoeffding’s test and naturally extends it to the two-sample setting. The main contributions include establishing the asymptotic optimality of the proposed test in both one-sample and two-sample scenarios and, for the first time, proving a strong converse theorem for the two-sample case. These results provide a unified and rigorous theoretical foundation for nonparametric hypothesis testing.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• one-sample test
• two-sample test
• binary hypothesis testing
• asymptotic optimality
• relative entropy

Innovation

• asymptotic optimality
• Hoeffding's test
• relative entropy
• two-sample testing
• strong converse