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Zonoids, linear dependence, and size-biased distributions on the simplex

  • Marco Dall'Aglio (a1) and Marco Scarsini (a2)
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.

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Corresponding author
Postal address: Dipartimento di Scienze, Università d'Annunzio, Viale Pindaro 42, I-65127 Pescara, Italy.
∗∗ Postal address: Dipartimento di Statistica e Matematica Applicata, Università di Torino, Piazza Arbarello 8, I-10122 Torino, Italy. Email address: marco.scarsini@unito.it
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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