On the Finite-Sample Performance of Measure Transportation-Based Multivariate Rank Tests

TL;DR

This paper investigates how the finite-sample performance of multivariate rank tests based on measure transportation is affected by the choice of grid, focusing on the two-sample location problem.

Contribution

It provides an analysis of the finite-sample effects of grid choice in measure transportation-based multivariate rank tests, extending univariate concepts to higher dimensions.

Findings

Finite-sample performance varies with grid choice.

Asymptotic properties are unaffected by grid selection.

Guidelines for grid selection in practice are suggested.

Abstract

Extending to dimension 2 and higher the dual univariate concepts of ranks and quantiles has remained an open problem for more than half a century. Based on measure transportation results, a solution has been proposed recently under the name center-outward ranks and quantiles which, contrary to previous proposals, enjoys all the properties that make univariate ranks a successful tool for statistical inference. Just as their univariate counterparts (to which they reduce in dimension one), center-outward ranks allow for the construction of distribution-free and asymptotically efficient tests for a variety of problems where the density of some noise or innovation remains unspecified. The actual implementation of these tests involves the somewhat arbitrary choice of a grid. While the asymptotic impact of that choice is nil, its finite-sample consequences are not. In this note, we investigate the finite-sample impact of that choice in the typical context of the multivariate two-sample location problem.