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Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

Author

Listed:
  • Victor Chernozhukov

    (MIT Department of Economics)

  • Alfred Galichon

    (Département d'économie)

  • Marc Hallin

    (Université Libre de Bruxelles (ULB))

  • Marc Henry

    (Départment de sciences économiques)

Abstract

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.

Suggested Citation

  • Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2015. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Sciences Po publications info:hdl:2441/3qnaslliat8, Sciences Po.
    • Handle: RePEc:spo:wpmain:info:hdl:2441/3qnaslliat80pbqa8t90240unj
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    References listed on IDEAS

    as
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    8. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
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    Citations

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    Cited by:

    1. Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    2. Kotík, Lukáš & Hlubinka, Daniel, 2017. "A weighted localization of halfspace depth and its properties," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 53-69.
    3. Marc Hallin & Davide La Vecchia & H Liu, 2019. "Center-Outward R-Estimation for Semiparametric VARMA Models," Working Papers ECARES 2019-25, ULB -- Universite Libre de Bruxelles.
    4. Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich towards a “General Generalised Distance”," Working Papers ECARES 2018-12, ULB -- Universite Libre de Bruxelles.
    5. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    6. Manuel Arellano & Stephane Bonhomme, 2019. "Recovering Latent Variables by Matching," Papers 1912.13081, arXiv.org.
    7. Florian Gunsilius, 2018. "Point-identification in multivariate nonseparable triangular models," Papers 1806.09680, arXiv.org.
    8. Eustasio Del Barrio & Juan Cuesta Albertos & Marc Hallin & Carlos Matran, 2018. "Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions," Working Papers ECARES 2018-15, ULB -- Universite Libre de Bruxelles.
    9. Florian Gunsilius & Susanne M. Schennach, 2019. "Independent nonlinear component analysis," CeMMAP working papers CWP46/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

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