Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

TL;DR

This paper introduces Monge-Kantorovich depth and related multivariate quantiles, ranks, and signs based on optimal transport maps, providing a flexible framework that captures complex distribution features beyond traditional methods.

Contribution

It develops a new depth concept using optimal transport, extending classical depth measures to non-convex and complex distributions, with consistent empirical estimators.

Findings

Monge-Kantorovich depth generalizes halfspace depth for spherical distributions.

Empirical transport maps converge uniformly, ensuring consistency of the proposed estimators.

The new framework captures non-convex features of distributions.

Abstract

We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on $R^d$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth in the case of spherical distributions, but, for more general distributions, differs from the latter in the ability for its contours to account for non convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge-Kantorovich depth contours, quantiles, ranks and signs, and show their consistency by establishing a uniform convergence property for empirical transport maps, which is of independent interest.