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Testing the equality of two conditional distributions is crucial in variousmodern applications, including transfer learning and causal inference. Despiteits importance, this fundamental problem has received surprisingly littleattention in the literature. This work aims to present a unified frameworkbased on distance and kernel methods for both global and local two-sampleconditional distribution testing. To this end, we introduce distance andkernel-based measures that characterize the homogeneity of two conditionaldistributions. Drawing from the concept of conditional U-statistics, we proposeconsistent estimators for these measures. Theoretically, we derive theconvergence rates and the asymptotic distributions of the estimators under boththe null and alternative hypotheses. Utilizing these measures, along with alocal bootstrap approach, we develop global and local tests that can detectdiscrepancies between two conditional distributions at global and local levels,respectively. Our tests demonstrate reliable performance through simulationsand real data analyses.

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