Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing

TL;DR

This paper introduces a unified framework using distance and kernel methods for both global and local two-sample conditional distribution testing, addressing a key gap in the literature.

Contribution

It presents the first unified approach for global and local testing of conditional distributions, including new measures, estimators, and theoretical analysis.

Findings

Reliable detection of distribution discrepancies demonstrated in simulations

Theoretical convergence rates and asymptotic distributions derived

Effective real data analysis confirms practical utility

Abstract

Testing the equality of two conditional distributions is crucial in various modern applications, including transfer learning and causal inference. Despite its importance, this fundamental problem has received surprisingly little attention in the literature, with existing works focusing exclusively on global two-sample conditional distribution testing. Based on distance and kernel methods, this paper presents the first unified framework for both global and local two-sample conditional distribution testing. To this end, we introduce distance and kernel-based measures that characterize the homogeneity of two conditional distributions. Drawing from the concept of conditional U-statistics, we propose consistent estimators for these measures. Theoretically, we derive the convergence rates and the asymptotic distributions of the estimators under both the null and alternative hypotheses. Utilizing these measures, along with a local bootstrap approach, we develop global and local tests that can detect discrepancies between two conditional distributions at global and local levels, respectively. Our tests demonstrate reliable performance through simulations and real data analysis.