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

Part of: Multivariate analysis Distribution theory - Probability General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Marco Dall'Aglio  and
Marco Scarsini
Marco Dall'Aglio*
Affiliation:
Università d'Annunzio, Pescara
Marco Scarsini*
Affiliation:
Università di Torino
*
∗ 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: [email protected]
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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.

Keywords

Zonoidzonotopelinear dependencecompositional variablesmultivariate size-biased distributionconcordance orderMarshall-Olkin distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60E15: Inequalities; stochastic orderings 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 871 - 884
Copyright
Copyright © Applied Probability Trust 2003 

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