Annals of Statistics
- Ann. Statist.
- Volume 45, Number 1 (2017), 223-256.
Monge–Kantorovich depth, quantiles, ranks and signs
Victor Chernozhukov, Alfred Galichon, Marc Hallin, and Marc Henry
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.
Article information
Source
Ann. Statist., Volume 45, Number 1 (2017), 223-256.
Dates
Received: January 2015
Revised: February 2016
First available in Project Euclid: 21 February 2017
Permanent link to this document
https://projecteuclid.org/euclid.aos/1487667622
Digital Object Identifier
doi:10.1214/16-AOS1450
Mathematical Reviews number (MathSciNet)
MR3611491
Zentralblatt MATH identifier
06710510
Subjects
Primary: 62M15: Spectral analysis 62G35: Robustness
Keywords
Statistical depth vector quantiles vector ranks multivariate signs empirical transport maps uniform convergence of empirical transport
Citation
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. https://projecteuclid.org/euclid.aos/1487667622
Supplemental materials
- Supplement to “Monge–Kantorovich depth, quantiles, ranks and signs”. In the online supplement [8], we provide a proof of Lemma 3.1.Digital Object Identifier: doi:10.1214/16-AOS1450SUPP