Open Access
February 2017 Monge–Kantorovich depth, quantiles, ranks and signs
Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry
Ann. Statist. 45(1): 223-256 (February 2017). DOI: 10.1214/16-AOS1450


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.


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Victor Chernozhukov. Alfred Galichon. Marc Hallin. Marc Henry. "Monge–Kantorovich depth, quantiles, ranks and signs." Ann. Statist. 45 (1) 223 - 256, February 2017.


Received: 1 January 2015; Revised: 1 February 2016; Published: February 2017
First available in Project Euclid: 21 February 2017

zbMATH: 06710510
MathSciNet: MR3611491
Digital Object Identifier: 10.1214/16-AOS1450

Primary: 62G35 , 62M15

Keywords: empirical transport maps , multivariate signs , Statistical depth , uniform convergence of empirical transport , vector quantiles , vector ranks

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 1 • February 2017
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