Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation

Understand

In this paper, 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.