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Monge-Kantorovich Depth, Quantiles, Ranks and Signs

Author

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  • Victor Chernozhukov
  • Alfred Galichon
  • Marc Hallin
  • Marc Henry

Abstract

We propose new concepts of statistical depth, multivariate quantiles,ranks and signs, based on canonical transportation maps between a distributionof interest on IRd and a reference distribution on the d-dimensionalunit ball. The new depth concept, called Monge-Kantorovich depth, specializesto halfspace depth in the case of elliptical distributions, but, for more generaldistributions, differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest. We propose empirical counterpartsto the population versions of those Monge-Kantorovich depth contours,quantiles, ranks and signs, and show their consistency by establishing a uniformconvergence property for transport maps, which is of independent interest.

Suggested Citation

  • Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2015. "Monge-Kantorovich Depth, Quantiles, Ranks and Signs," Working Papers ECARES ECARES 2015-02, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/190592
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    Keywords

    statictical depth; vector quantiles; vector ranks; multivariate signs; optimal transport maps;
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