  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 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 and signs, and show their consistency by
establishing a uniform convergence property for empirical transport maps, which
is of independent interest.
