Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

Victor Chernozhukov, Alfred Galichon (), Marc Hallin and Marc Henry
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Alfred Galichon: ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique

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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.

Keywords: statistical depth; quantiles; ranks; signs (search for similar items in EconPapers)
Date: 2017
Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-03391975
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Citations: View citations in EconPapers (32)

Published in Annals of Statistics, 2017, 45 (1), pp.223 - 256. ⟨10.1214/16-AOS1450⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-03391975

DOI: 10.1214/16-AOS1450

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