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Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex

Author

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  • Marco Scarsini

    (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique, Dipartimento di Scienze Economiche e Aziendali - LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma])

  • Marco Dall'Aglio

Abstract

The zonoid of a d-dimensional random vector is used as a tool for measuring linear dependence among its components. A preorder of linear dependence is defined through inclusion of the zonoids. The zonoid of a random vector does not characterize its distribution, but it does characterize the size-biased distribution of its compositional variables. This fact will allow a characterization of our linear dependence order in terms of a linear-convex order for the size-biased compositional variables. In dimension 2 the linear dependence preorder will be shown to be weaker than the concordance order. Some examples related to the Marshall-Olkin distribution and to a copula model will be presented, and a class of measures of linear dependence will be proposed.

Suggested Citation

  • Marco Scarsini & Marco Dall'Aglio, 2003. "Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex," Post-Print hal-00539799, HAL.
    • Handle: RePEc:hal:journl:hal-00539799
    DOI: 10.1239/aap/1067436324
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    References listed on IDEAS

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    Cited by:

    1. Taizhong Hu & Alfred Müller & Marco Scarsini, 2002. "Some Counterexamples in Positive Dependence," ICER Working Papers - Applied Mathematics Series 28-2003, ICER - International Centre for Economic Research, revised Jul 2003.
    2. Colangelo, Antonio & Scarsini, Marco & Shaked, Moshe, 2006. "Some positive dependence stochastic orders," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 46-78, January.
    3. Paolo Giudici & Emanuela Raffinetti, 2020. "Lorenz Model Selection," Journal of Classification, Springer;The Classification Society, vol. 37(3), pages 754-768, October.
    4. Müller, Alfred & Scarsini, Marco, 2005. "Archimedean copulæ and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 434-445, April.
    5. Marc-Arthur Diaye & Gleb Koshevoy & Ilya Molchanov, 2019. "Lift expectations of random sets [Augmenter les attentes concernant les ensembles aléatoires]," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03897964, HAL.
    6. Diaye, Marc-Arthur & Koshevoy, Gleb A. & Molchanov, Ilya, 2019. "Lift expectations of random sets," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 110-117.
    7. Antonio Lijoi & Igor Prünster & Stephen G. Walker, 2004. "On consistency of nonparametric normal mixtures for Bayesian density estimation," ICER Working Papers - Applied Mathematics Series 23-2004, ICER - International Centre for Economic Research.
    8. Antonio Lijoi & Igor Prünster & Stephen G. Walker, 2004. "On rates of convergence for posterior distributions in infinite–dimensional models," ICER Working Papers - Applied Mathematics Series 24-2004, ICER - International Centre for Economic Research.
    9. Antonio Lijoi & Igor Prünster & Stephen G. Walker, 2004. "Contributions to the understanding of Bayesian consistency," ICER Working Papers - Applied Mathematics Series 13-2004, ICER - International Centre for Economic Research.
    10. Marc-Arthur Diaye & Gleb Koshevoy & Ilya Molchanov, 2019. "Lift expectations of random sets [Augmenter les attentes concernant les ensembles aléatoires]," Post-Print hal-03897964, HAL.

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