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Center-Outward Quantiles And The Measurement Of Multivariate Risk

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  • Jan Bierlant
  • Sven Buitendag
  • Eustasio Del Barrio
  • Marc Hallin

Abstract

All multivariate extensions of the univariate theory of risk measurement run into the same fundamental problem of the absence, in dimension d > 1, of a canonical ordering of Rd. Based on measure transportation ideas, several attempts have been made recently in the statistical literature to overcome that conceptual difficulty. In Hallin (2017), the concepts of center-outward distribution and quantile functions are developed as generalisations of the classical univariate concepts of distribution and quantile functions, along with their empirical versions. The center-outward distribution function F± is a homeomorphic cyclically monotone mapping from Rd \ F−1 ± (0) to the open punctured unit ball Bd \ {0}, while its empirical counterpart F(n) ± is a cyclically monotone mapping from the sample to a regular grid over Bd. In dimension d = 1, F± reduces to 2F − 1, while F(n) ± generates the same sigma-field as traditional univariate ranks. The empirical F(n) ± ,however, involves a large number of ties, which is impractical in the context of risk measurement. We therefore propose a class of smooth approximations Fn,ξ (ξ a smoothness index) of F(n) ± as an alternative to the interpolation developed in del Barrio et al. (2018). This approximation allows for the computation of some new empirical risk measures, based either on the convex potential associated with the proposed transports, or on the volumes of the resulting empirical quantile regions. We also discuss the role of such transports in the evaluation of the risk associated with multivariate regularly varying distributions. Some simulations and applications to case studies illustrate the value of the approach.

Suggested Citation

  • Jan Bierlant & Sven Buitendag & Eustasio Del Barrio & Marc Hallin, 2019. "Center-Outward Quantiles And The Measurement Of Multivariate Risk," Working Papers ECARES 2019-30, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/297778
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    References listed on IDEAS

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    1. Rahul Mazumder & Arkopal Choudhury & Garud Iyengar & Bodhisattva Sen, 2019. "A Computational Framework for Multivariate Convex Regression and Its Variants," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(525), pages 318-331, January.
    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:dau:papers:123456789/2278 is not listed on IDEAS
    4. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    5. Alfred Galichon & Ivar Ekeland & Marc Henry, 2009. "Comonotonic measures of multivariates risks," Working Papers hal-00401828, HAL.
    6. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
    7. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    8. Beirlant, J. & Mason, D. M. & Vynckier, C., 1999. "Goodness-of-fit analysis for multivariate normality based on generalized quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 30(2), pages 119-142, April.
    9. 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.
    10. Einmahl, J. H.J. & Mason, D.M., 1992. "Generalized quantile processes," Other publications TiSEM b2a76bac-045d-457f-869f-d, Tilburg University, School of Economics and Management.
    11. A. Dematteo & S. Clémençon, 2016. "On tail index estimation based on multivariate data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 152-176, March.
    12. 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.
    13. Arthur Charpentier, 2018. "An introduction to multivariate and dynamic risk measures," Working Papers hal-01831481, HAL.
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    Cited by:

    1. Eustasio Del Barrio & Alberto Gonzalez-Sanz & Marc Hallin, 2019. "A Note on the Regularity of Center-Outward Distribution and Quantile Functions," Working Papers ECARES 2019-33, ULB -- Universite Libre de Bruxelles.
    2. Segers, Johan, 2022. "Graphical and uniform consistency of estimated optimal transport plans," LIDAM Discussion Papers ISBA 2022022, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Hongjian Shi & Marc Hallin & Mathias Drton & Fang Han, 2020. "Rate-Optimality of Consistent Distribution-Free Tests of Independence Based on Center-Outward Ranks and Signs," Working Papers ECARES 2020-23, ULB -- Universite Libre de Bruxelles.
    4. 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.
    5. 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).
    6. 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.
    7. Marc Hallin, 2021. "Measure Transportation and Statistical Decision Theory," Working Papers ECARES 2021-04, ULB -- Universite Libre de Bruxelles.
    8. 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.

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