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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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    References listed on IDEAS

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    15. 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.
    16. Davy Paindaveine & Miroslav Šiman, 2012. "Computing multiple-output regression quantile regions from projection quantiles," Computational Statistics, Springer, vol. 27(1), pages 29-49, March.
    17. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    18. Anil K. Ghosh & Probal Chaudhuri, 2005. "On Maximum Depth and Related Classifiers," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 327-350, June.
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    Cited by:

    1. Yanqin Fan & Marc Henry, 2020. "Vector copulas," Papers 2009.06558, arXiv.org, revised Apr 2021.
    2. Dmitry Arkhangelsky, 2019. "Dealing with a Technological Bias: The Difference-in-Difference Approach," Working Papers wp2019_1903, CEMFI.
    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 & Gilles Mordant, 2021. "On the Finite-Sample Performance of Measure Transportation-Based Multivariate Rank Tests," Working Papers ECARES 2021-24, ULB -- Universite Libre de Bruxelles.
    5. 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.
    6. 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.
    7. Marc Hallin & Daniel Hlubinka & Sarka Hudecova, 2020. "Fully Distribution-free Center-outward Rank Tests for Multiple-output Regression and Manova," Working Papers ECARES 2020-32, ULB -- Universite Libre de Bruxelles.
    8. Marc Hallin & Davide La Vecchia & Hang Liu, 2020. "Rank-Based Testing for Semiparametric VAR Models: a measure transportation approach," Working Papers ECARES 2020-47, ULB -- Universite Libre de Bruxelles.
    9. 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.
    10. Jonas Meier, 2020. "Multivariate Distribution Regression," Diskussionsschriften dp2023, Universitaet Bern, Departement Volkswirtschaft.
    11. 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.
    12. Manuel Arellano & Stéphane Bonhomme, 2019. "Recovering Latent Variables by Matching," Working Papers wp2019_1914, CEMFI.
    13. Marc Hallin & Hongjian Shi & Mathias Drton & Fang Han, 2021. "Center-Outward Sign- and Rank-Based Quadrant, Spearman, and Kendall Tests for Multivariate Independence," Working Papers ECARES 2021-27, ULB -- Universite Libre de Bruxelles.
    14. 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.
    15. María Edo & Walter Sosa Escudero & Marcela Svarc, 2021. "A multidimensional approach to measuring the middle class," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 19(1), pages 139-162, March.
    16. Lixiong Li & Marc Henry, 2022. "Finite Sample Inference in Incomplete Models," Papers 2204.00473, arXiv.org.
    17. Marc Hallin, 2021. "Measure Transportation and Statistical Decision Theory," Working Papers ECARES 2021-04, ULB -- Universite Libre de Bruxelles.
    18. Marc Hallin & Daniel Hlubinka & Sarka Hudecova, 2021. "Efficient Fully Distribution-Free Center-Outward Rank Tests for Multiple-Output Regression and MANOVA," Working Papers ECARES 2021-13, ULB -- Universite Libre de Bruxelles.
    19. Florian Gunsilius, 2018. "Point-identification in multivariate nonseparable triangular models," Papers 1806.09680, arXiv.org.
    20. Alfred Galichon & Bernard Salani'e, 2021. "Cupid's Invisible Hand: Social Surplus and Identification in Matching Models," Papers 2106.02371, arXiv.org.
    21. Yanqin Fan & Marc Henry & Brendan Pass & Jorge A. Rivero, 2022. "Lorenz map, inequality ordering and curves based on multidimensional rearrangements," Papers 2203.09000, arXiv.org.
    22. Faugeras, Olivier P. & Rüschendorf, Ludger, 2021. "Functional, randomized and smoothed multivariate quantile regions," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    23. Alfred Galichon, 2021. "The unreasonable effectiveness of optimal transport in economics," Papers 2107.04700, arXiv.org.
    24. 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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    Keywords

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