Monge-kantorovich depth, quantiles, ranks and signs

Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry

Research output: Contribution to journalArticlepeer-review

99 Scopus citations


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
Issue number1
StatePublished - Feb 2017

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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