Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation

Abstract

In this article, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that are exactly distribution-free under the null hypothesis. We demonstrate the applicability of this approach by constructing exact distribution-free tests for two classical nonparametric problems: (I) testing for mutual independence between random vectors, and (II) testing for the equality of multivariate distributions. In particular, we propose (multivariate) rank versions of distance covariance and energy statistic for testing scenarios (I) and (II), respectively. In both these problems, we derive the asymptotic null distribution of the proposed test statistics. We further show that our tests are consistent against all fixed alternatives. Moreover, the proposed tests are computationally feasible and are well-defined under minimal assumptions on the underlying distributions (e.g., they do not need any moment assumptions). We also demonstrate the efficacy of these procedures via extensive simulations. In the process of analyzing the theoretical properties of our procedures, we end up proving some new results in the theory of measure transportation and in the limit theory of permutation statistics using Stein’s method for exchangeable pairs, which may be of independent interest.

Keywords:

• Asymptotic null distribution
• Consistency against fixed alternatives
• Distance covariance
• Distribution-free inference
• Energy distance
• Multivariate ranks
• Multivariate two-sample testing
• Quasi-Monte Carlo sequences
• Stein’s method for exchangeable pairs
• Testing for mutual independence

Supplementary Materials

The supplementary material, which is available online, contains proofs of our main results, real data experiments and additional computational studies.

Acknowledgments

We would like to thank the associate editor and the two anonymous reviewers for their constructive comments that helped improve the quality of this article.

Additional information

Funding

This work was supported by NSF (grant no. DMS-2015376).