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Functional, randomized and smoothed multivariate quantile regions

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  • Faugeras, Olivier
  • Rüschendorf, Ludger

Abstract

A notion of multivariate depth, resp. quantile region, was introduced in [Chernozhukov et al., 2017], based on a mass transportation approach. In [Faugeras and Ruschendorf, 2017], this approach was generalized by dening quantiles as Markov morphisms carrying suitable algebraic, ordering and topological structures over probability measures. In addition, a copula step was added to the mass transportation step. Empirical versions of these depth areas do not give exact level depth regions. In this paper, we introduce randomized depth regions by means of a formulation by depth functions, resp. by randomized quantiles sets. These versions attain the exact level and also provide the corresponding consistency property. We also investigate in the case of continuous marginals a smoothed version of the empirical copula and compare its behavior with the unsmoothed version. Extensive simulations illustrate the resulting randomized depth areas and show that they give a valid representation of the central depth areas of a multivariate distribution, and thus are a valuable tool for their analysis.

Suggested Citation

  • Faugeras, Olivier & Rüschendorf, Ludger, 2019. "Functional, randomized and smoothed multivariate quantile regions," TSE Working Papers 19-1039, Toulouse School of Economics (TSE), revised Jun 2021.
    • Handle: RePEc:tse:wpaper:123569
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    1. Cuesta-Albertos, J. A. & Matrán, C. & Tuero-Diaz, A., 1997. "Optimal Transportation Plans and Convergence in Distribution," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 72-83, January.
    2. Yukich, J. E., 1989. "A note on limit theorems for perturbed empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 33(1), pages 163-173, October.
    3. Faugeras Olivier P., 2017. "Inference for copula modeling of discrete data: a cautionary tale and some facts," Dependence Modeling, De Gruyter, vol. 5(1), pages 121-132, January.
    4. Marc Hallin, 2017. "On Distribution and Quantile Functions, Ranks and Signs in R_d," Working Papers ECARES ECARES 2017-34, ULB -- Universite Libre de Bruxelles.
    5. Rüschendorf, L. & Rachev, S. T., 1990. "A characterization of random variables with minimum L2-distance," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 48-54, January.
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    1. 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).
    2. 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.

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