Monge-kantorovich depth, quantiles, ranks and signs

Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry

    Research output: Contribution to journalArticle


    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 1 2017



    • Empirical transport maps
    • Multivariate signs
    • Statistical depth
    • Uniform convergence of empirical transport
    • Vector quantiles
    • Vector ranks

    ASJC Scopus subject areas

    • Statistics and Probability
    • Statistics, Probability and Uncertainty

    Cite this

    Chernozhukov, V., Galichon, A., Hallin, M., & Henry, M. (2017). Monge-kantorovich depth, quantiles, ranks and signs. Annals of Statistics, 45(1), 223-256.