Multivariate Brenier cumulative distribution functions and their application to non-parametric testing

Abstract

In this work we introduce a novel approach of construction of multivariate cumulative distribution functions, based on cyclical-monotone mapping of an original measure $\mu \in {\mathcal{P}}_2^{ac}({\mathbb{R}}^d)$ to some target measure $\nu \in {\mathcal{P}}_2^{ac}({\mathbb{R}}^d)$ , supported on a convex compact subset of ${\mathbb{R}}^d$. This map is referred to as $\nu$-Brenier distribution function ($\nu$-BDF), whose counterpart under the one-dimensional setting $d=1$ is an ordinary CDF, with $\nu$ selected as $\mathcal{U}[0,1]$, a uniform distribution on $[0,1]$. Following one-dimensional frame-work, a multivariate analogue of Glivenko-Cantelli theorem is provided. A practical applicability of the theory is then illustrated by the development of a non-parametric pivotal two-sample test, that is rested on $2$-Wasserstein distance.

arXiv comment19 pages, 2 figures

References in corpus (1)

• Nonparametric multivariate rank tests and their unbiasedness

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