Monge-kantorovich depth, quantiles, ranks and signs

Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry

Research output: Contribution to journalArticle

19 Citations (Scopus)

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 Rd 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.

Original languageEnglish (US)
Pages (from-to)223-256
Number of pages34
JournalAnnals of Statistics
Volume45
Issue number1
DOIs
StatePublished - Feb 2017

Fingerprint

Quantile
Halfspace Depth
Spherical Distribution
Uniform convergence
Unit ball
Convergence Properties
Reverse
Concepts

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Chernozhukov, Victor ; Galichon, Alfred ; Hallin, Marc ; Henry, Marc. / Monge-kantorovich depth, quantiles, ranks and signs. In: Annals of Statistics. 2017 ; Vol. 45, No. 1. pp. 223-256.
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Chernozhukov, V, Galichon, A, Hallin, M & Henry, M 2017, 'Monge-kantorovich depth, quantiles, ranks and signs', Annals of Statistics, vol. 45, no. 1, pp. 223-256. https://doi.org/10.1214/16-AOS1450

Monge-kantorovich depth, quantiles, ranks and signs. / Chernozhukov, Victor; Galichon, Alfred; Hallin, Marc; Henry, Marc.

In: Annals of Statistics, Vol. 45, No. 1, 02.2017, p. 223-256.

Research output: Contribution to journalArticle

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