A Semi-Supervised Kernel Two-Sample Test

TL;DR

This paper introduces a semi-supervised kernel two-sample test that leverages covariate data to improve testing power while maintaining proper calibration through asymptotic normality.

Contribution

It proposes a novel semi-supervised test statistic with asymptotic normality, enhancing power and consistency in two-sample testing with covariates.

Findings

Test achieves higher asymptotic power than existing kernel tests without covariates.

The proposed method is consistent against fixed and local alternatives.

Simulations validate the theoretical advantages of the approach.

Abstract

We consider the problem of two-sample testing in a semi-supervised setting with abundant unlabeled covariate data. Standard two-sample tests neglect covariate information, which has the potential to significantly boost performance. However, incorporating covariates potentially breaks the exchangeability assumption under the null, which further complicates a calibration procedure. To address these issues, we propose a semi-supervised method that produces a test statistic with asymptotic normality, while effectively integrating additional information from covariates. Our test is straightforward to calibrate due to the asymptotic normality under the null and achieves asymptotic power that is often much higher than existing kernel tests without covariates. Furthermore, we formally show that the proposed method is consistent in power against fixed and local alternatives. Simulations confirm the practical and theoretical strengths of our approach.