Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

Victor Chernozhukov , Alfred Galichon (1) , Marc Hallin (2) , Marc Henry (3, 4)

1 ECON - Département d'économie (Sciences Po)
2 ULB - Université libre de Bruxelles
3 CIRANO - Centre interuniversitaire de recherche en analyse des organisations
4 CIREQ - Centre interuniversitaire de recherche en économie quantitative

We propose new concepts of statistical depth, multivariate quantiles, 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 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 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.

Mots clés: Statistical Depth; Vector Quantiles; Vector Ranks; Multivariate Signs; Empirical Transport Maps; Uniform Convergence of Empirical Transport

Domaines: Economies et finances

Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry. Monge-Kantorovich Depth, Quantiles, Ranks, and Signs. 2015. ⟨hal-03460056⟩