The Annals of Statistics

Monge–Kantorovich depth, quantiles, ranks and signs

Victor Chernozhukov, Alfred Galichon, Marc Hallin, and Marc Henry

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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.

Article information

Ann. Statist., Volume 45, Number 1 (2017), 223-256.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M15: Spectral analysis 62G35: Robustness

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


Chernozhukov, Victor; Galichon, Alfred; Hallin, Marc; Henry, Marc. Monge–Kantorovich depth, quantiles, ranks and signs. Ann. Statist. 45 (2017), no. 1, 223--256. doi:10.1214/16-AOS1450.

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Supplemental materials

  • Supplement to “Monge–Kantorovich depth, quantiles, ranks and signs”. In the online supplement [8], we provide a proof of Lemma 3.1.