Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-21T13:58:01.840Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: marco.scarsini@unito.it
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bolker, E. D. (1969). A class of convex bodies. Trans. Amer. Math. Soc. 145, 323–345.CrossRefGoogle Scholar
Chacon, R. V. and Walsh, J. B. (1976). One-dimensional potential embedding. In Séminaire de Probabilités, X (Notes Math. 511), Springer, Berlin, pp. 19–23.Google Scholar
Choquet, G. (1969). Mesures coniques, affines et cylindriques. In Symposia Mathematica (INDAM, Rome, 1968), Vol. II, Academic Press, London, pp. 145–182.Google Scholar
Dall'Aglio, M. and Scarsini, M. (2001). When Lorenz met Lyapunov. Statist. Prob. Lett. 54, 101–105.Google Scholar
Elton, J. and Hill, T. P. (1992). Fusions of a probability distribution. Ann. Prob. 20, 421–454.CrossRefGoogle Scholar
Gini, C. (1914). Sulla misura della concentrazione e della variabilità dei caratteri. Atti R. Ist. Veneto Sci. Lett. Arti 73, 185–213.Google Scholar
Goodey, P. and Weil, W. (1993). Zonoids and generalisations. In Handbook of Convex Geometry, Vol. B, North-Holland, Amsterdam, pp. 1297–1326.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
Kimeldorf, G. and Sampson, A. R. (1987). Positive dependence orderings. Ann. Inst. Statist. Math. 39, 113–128.Google Scholar
Kimeldorf, G. and Sampson, A. R. (1989). A framework for positive dependence. Ann. Inst. Statist. Math. 41, 31–45.Google Scholar
Koshevoy, G. and Mosler, K. (1996). The Lorenz zonoid of a multivariate distribution. J. Amer. Statist. Assoc. 91, 873–882.Google Scholar
Koshevoy, G. and Mosler, K. (1997). Multivariate Gini indices. J. Multivariate Anal. 60, 252–276.Google Scholar
Koshevoy, G. and Mosler, K. (1998). Lift zonoids, random convex hulls and the variability of random vectors. Bernoulli 4, 377–399.CrossRefGoogle Scholar
Machina, M. J. and Pratt, J. W. (1997). Increasing risk: some direct constructions. J. Risk Uncertainty 14, 103–127.Google Scholar
Marshall, A. W. and Olkin, I. (1967). A generalized bivariate exponential distribution. J. Appl. Prob. 4, 291–302.Google Scholar
Mosler, K. (2002). Multivariate Dispersion, Central Regions and Depth (Lecture Notes Statist. 165). Springer, Berlin.Google Scholar
Müller, A., (1997). Stochastic orders generated by integrals: a unified study. Adv. Appl. Prob. 29, 414–428.Google Scholar
Rothschild, M. and Stiglitz, J. E. (1970). Increasing risk. I. A definition. 2, 225–243.Google Scholar
Scarsini, M. (1998). Multivariate convex orderings, dependence, and stochastic equality. J. Appl. Prob. 35, 93–103.Google Scholar
Scarsini, M. and Shaked, M. (1990). Some conditions for stochastic equality. Naval Res. Logistics 37, 617–625.Google Scholar
Scarsini, M. and Shaked, M. (1996). Positive dependence orders: a survey. In Athens Conference on Applied Probability and Time Series Analysis (Lecture Notes Statist. 114), Vol. I, Springer, New York, pp. 70–91.CrossRefGoogle Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Tchen, A. H. (1980). Inequalities for distributions with given marginals. Ann. Prob. 8, 814–827.Google Scholar