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

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Author Info

  • Marco Dall’Aglio

    ()

  • Marco Scarsini

    ()

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 characterizes 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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File URL: http://www.icer.it/docs/wp2003/Scarsini27-03.pdf
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Bibliographic Info

Paper provided by ICER - International Centre for Economic Research in its series ICER Working Papers - Applied Mathematics Series with number 27-2003.

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Length: 25 pages
Date of creation: Dec 2000
Date of revision: Jul 2003
Handle: RePEc:icr:wpmath:27-2003

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Related research

Keywords: zonoid; zonotope; linear dependence; compositional variables; multivariate size biased distribution; concordance order; Marshall-Olkin distribution.;

References

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  1. Machina, Mark J & Pratt, John W, 1997. "Increasing Risk: Some Direct Constructions," Journal of Risk and Uncertainty, Springer, vol. 14(2), pages 103-27, March.
  2. Gleb Koshevoy, 1997. "The Lorenz zonotope and multivariate majorizations," Social Choice and Welfare, Springer, vol. 15(1), pages 1-14.
  3. Koshevoy, Gleb & Mosler, Karl, 1995. "Multivariate Gini indices," Discussion Papers in Statistics and Econometrics 7/95, University of Cologne, Department for Economic and Social Statistics.
  4. K. Mosler, 2003. "Central regions and dependency," Econometrics 0309004, EconWPA.
  5. Dall'Aglio, Marco & Scarsini, Marco, 2001. "When Lorenz met Lyapunov," Statistics & Probability Letters, Elsevier, vol. 54(1), pages 101-105, August.
  6. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
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Cited by:
  1. 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.
  2. 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.
  3. 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.
  4. Müller, Alfred & Scarsini, Marco, 2005. "Archimedean copulæ and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 434-445, April.
  5. 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.

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