%0 Journal Article %T Monge-Kantorovich Depth, Quantiles, Ranks, and Signs %+ Département d'économie (Sciences Po) (ECON) %+ Université libre de Bruxelles (ULB) %+ Centre interuniversitaire de recherche en analyse des organisations (CIRANO) %+ Centre interuniversitaire de recherche en économie quantitative (CIREQ) %A Chernozhukov, Victor %A Galichon, Alfred %A Hallin, Marc %A Henry, Marc %< avec comité de lecture %Z 2441/6no1u8c7jl8pqpcudqn9e5flms %@ 0090-5364 %J Annals of Statistics %I Institute of Mathematical Statistics %V 45 %N 1 %P 223 - 256 %8 2017 %D 2017 %R 10.1214/16-AOS1450 %K statistical depth %K quantiles %K ranks %K signs %Z Humanities and Social Sciences/Economics and FinanceJournal articles %X 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 thedistribution 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) transportmaps, which is the main theoretical result of this paper. %G English %2 https://sciencespo.hal.science/hal-03391975/document %2 https://sciencespo.hal.science/hal-03391975/file/2017-chernozhukov-galichon-et-alii-monge-kantorovich-depth-quantiles-ranks-and-signs.pdf %L hal-03391975 %U https://sciencespo.hal.science/hal-03391975 %~ SHS %~ SCIENCESPO %~ CNRS %~ AO-ECONOMIE %~ OPENAIRE %~ USPC %~ ECON-SCPO %~ SCPO_OA %~ SCPO-REPEC