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Monge-Kantorovich Depth, Quantiles, Ranks, and Signs

Author

Listed:
  • Victor Chernozhukov
  • Alfred Galichon

    (ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique)

  • Marc Hallin

    (ULB - Université libre de Bruxelles)

  • Marc Henry

    (CIRANO - Centre interuniversitaire de recherche en analyse des organisations - UQAM - Université du Québec à Montréal = University of Québec in Montréal, CIREQ - Centre interuniversitaire de recherche en économie quantitative)

Abstract

We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, 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 for d = 1 and in the case of spherical 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, signs and vector quantiles and ranks, and show their consistency by establishing a uniform convergence property for empirical (forward and reverse) transport maps, which is the main theoretical result of this paper.

Suggested Citation

  • Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2017. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Post-Print hal-03391975, HAL.
  • Handle: RePEc:hal:journl:hal-03391975
    DOI: 10.1214/16-AOS1450
    Note: View the original document on HAL open archive server: https://hal-sciencespo.archives-ouvertes.fr/hal-03391975
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    References listed on IDEAS

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    1. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0p30p95 is not listed on IDEAS
    2. Hassairi, Abdelhamid & Regaieg, Ons, 2008. "On the Tukey depth of a continuous probability distribution," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2308-2313, October.
    3. Hallin, Marc & Paindaveine, Davy, 2005. "Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 122-163, March.
    4. Davy Paindaveine & Germain Van bever, 2013. "From Depth to Local Depth: A Focus on Centrality," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 1105-1119, September.
    5. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2014. "Vector quantile regression," CeMMAP working papers CWP48/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. Alfred Galichon & Marc Henry, 2012. "Dual theory of choice under multivariate risks," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    7. repec:dau:papers:123456789/2278 is not listed on IDEAS
    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.
    9. Alfred Galichon & Ivar Ekeland & Marc Henry, 2009. "Comonotonic measures of multivariates risks," Working Papers hal-00401828, HAL.
    10. Davy Paindaveine & Miroslav Šiman, 2012. "Computing multiple-output regression quantile regions from projection quantiles," Computational Statistics, Springer, vol. 27(1), pages 29-49, March.
    11. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2015. "Vector quantile regression: an optimal transport approach," CeMMAP working papers CWP58/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Karl Mosler, 2003. "Central Regions and Dependency," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 5-21, March.
    13. Koshevoy, Gleb A., 2002. "The Tukey Depth Characterizes the Atomic Measure," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 360-364, November.
    14. Alfred Galichon & Ivar Ekeland & Marc Henry, 2009. "Comonotonic measures of multivariates risks," Working Papers hal-00401828, HAL.
    15. repec:hal:wpspec:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
    16. Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232, May.
    17. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    18. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, December.
    19. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    20. Galichon, Alfred & Henry, Marc, 2012. "Dual theory of choice with multivariate risks," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1501-1516.
    21. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
    22. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    23. Alfred Galichon & Marc Henry, 2012. "Dual theory of choice under multivariate risks," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    24. repec:hal:spmain:info:hdl:2441/4c5431jp6o888pdrcs0fuirl40 is not listed on IDEAS
    25. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2016. "Vector Quantile Regression: An Optimal Transport Approach," SciencePo Working papers hal-03567920, HAL.
    26. 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.
    27. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
    28. Paindaveine, Davy & Šiman, Miroslav, 2012. "Computing multiple-output regression quantile regions," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 840-853.
    29. repec:hal:wpspec:info:hdl:2441/5rkqqmvrn4tl22s9mc0p30p95 is not listed on IDEAS
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