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Monge-Kantorovich depth, quantiles, ranks and signs

Author

Listed:
  • Victor Chernozhukov

    (Institute for Fiscal Studies and MIT)

  • Alfred Galichon

    (Institute for Fiscal Studies and NYU)

  • Marc Hallin

    (Institute for Fiscal Studies and Université Libre de Bruxelles)

  • Marc Henry

    (Institute for Fiscal Studies and Universite de Montreal)

Abstract

We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution 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 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.

Suggested Citation

  • Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2015. "Monge-Kantorovich depth, quantiles, ranks and signs," CeMMAP working papers CWP04/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:04/15
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