MONGE-KANTOROVICH DEPTH, QUANTILES, RANKS AND SIGNS

Abstract. We propose new concepts of statistical depth, multivariate quan-tiles, ranks and signs, based on canonical transportation maps between a dis-tribution 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 coun-terparts 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 in-terest