Monge–Kantorovich depth, quantiles, ranks and signs
Author(s)Galichon, Alfred; Hallin, Marc; Henry, Marc; Chernozhukov, Victor V
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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.
DepartmentMassachusetts Institute of Technology. Department of Economics
The Annals of Statistics
Institute of Mathematical Statistics
Chernozhukov, Victor, Alfred Galichon, Marc Hallin, and Marc Henry. “Monge–Kantorovich Depth, Quantiles, Ranks and Signs.” The Annals of Statistics 45, no. 1 (February 2017): 223–256.