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Measure Transportation and Statistical Decision Theory

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  • Marc Hallin

Abstract

Unlike the real line, the real space, in dimension $dgeq 2$, is not canonically ordered. As a consequence, extending to a multivariate context fundamental univariate statistical tools such as quantiles, signs, and ranks is anything but obvious. Tentative definitions have been proposed in the literature but do not enjoy the basic properties (e.g. distribution-freeness of ranks, their independence with respect to the order statistic, their independence with respect to signs, etc.) they are expected to satisfy. Based on measure transportation ideas, new concepts of distribution and quantile functions, ranks, and signs have been proposed recently that, unlike previous attempts, do satisfy these properties. These ranks, signs, and quantiles have been used, quite successfully, in several inference problems and have triggered, in a short span of time, a number of applications: fully distribution-free testing for multiple-output regression, MANOVA, and VAR models, R-estimation for VARMA parameters, distribution-free testing for vector independence, multiple-output quantile regression, nonlinear independent component analysis, etc.

Suggested Citation

  • Marc Hallin, 2021. "Measure Transportation and Statistical Decision Theory," Working Papers ECARES 2021-04, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/318373
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    References listed on IDEAS

    as
    1. Alfred Galichon & Ivar Ekeland & Marc Henry, 2009. "Comonotonic measures of multivariates risks," Working Papers hal-00401828, HAL.
    2. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    3. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
    4. 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.
    5. Beirlant, J. & Buitendag, S. & del Barrio, E. & Hallin, M. & Kamper, F., 2020. "Center-outward quantiles and the measurement of multivariate risk," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 79-100.
    6. repec:dau:papers:123456789/2278 is not listed on IDEAS
    7. 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.
    8. del Barrio, Eustasio & González-Sanz, Alberto & Hallin, Marc, 2020. "A note on the regularity of optimal-transport-based center-outward distribution and quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    9. Carlier, Guillaume & Chernozhukov, Victor & Galichon, Alfred, 2017. "Vector quantile regression beyond the specified case," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 96-102.
    10. Bakirov, Nail K. & Rizzo, Maria L. & Szekely, Gábor J., 2006. "A multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1742-1756, September.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Measure transportation; statistical decision theory;

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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