Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

Résumé

We propose new concepts of statistical depth, multivariate quantiles, ranks, and signs, based on canonical transportation maps between a distribution of interest on R^d 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. Throughout, emphasis will be put on the Monge-Kantorovich ranks and signs as distribution-free and semiparametrically efficient inferential tools.