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Monge-Kantorovich depth, quantiles, ranks and signs

Authors: Victor Chernozhukov , Alfred Galichon , Marc Hallin and Marc Henry
Date: 28 January 2015
Type: cemmap Working Paper, CWP04/15
DOI: 10.1920/wp.cem.2015.0415


We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on IRd and a reference distribution on the d-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth in the case of elliptical 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 and signs, and show their consistency by establishing a uniform convergence property for empirical transport maps, which is of independent interest.

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Victor Chernozhukov, Alfred Galichon, Marc Hallin and Marc Henry September 2015, Monge-Kantorovich depth, quantiles, ranks and signs, cemmap Working Paper, CWP57/15, Institute for Fiscal Studies

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