Monge–Kantorovich depth, quantiles, ranks and signs

Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry

The Annals of Statistics · 2017 · Volume 45 · Issue 1

Abstract

We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, ranks and signs, based on canonical transportation maps between a distribution of interest on $\mathbb{R}^{d}$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge–Kantorovich depth, specializes to halfspace depth for $d=1$ and 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, signs and vector quantiles and ranks, and show their consistency by establishing a uniform convergence property for empirical (forward and reverse) transport maps, which is the main theoretical result of this paper.

Keywords

62G35, 62M15, empirical transport maps, multivariate signs, Statistical depth, uniform convergence of empirical transport, vector quantiles, vector ranks

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