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

09/11/2018

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by   Melf Boeckel, et al.

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In this work we introduce a novel approach of construction of multivariate cumulative distribution functions, based on cyclical-monotone mapping of an original measure μ∈P^ac_2(R^d) to some target measure ν∈P^ac_2(R^d) , supported on a convex compact subset of R^d. This map is referred to as ν-Brenier distribution function (ν-BDF), whose counterpart under the one-dimensional setting d = 1 is an ordinary CDF, with ν selected as 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.