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Multivariate extremes, aggregation and dependence in elliptical distributions

Part of: Distribution theory - Probability Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

Henrik Hult  and
Filip Lindskog
Henrik Hult*
Affiliation:
KTH, Stockholm
Filip Lindskog*
Affiliation:
ETH, Zürich
*
∗ Postal address: Department of Mathematics, KTH, S-100 44 Stockholm, Sweden.
∗∗ Postal address: RiskLab, Department of Mathematics, ETH Zentrum, CH-8092 Zürich, Switzerland. Email address: [email protected]
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Abstract

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

Keywords

Elliptical distributionmultivariate extremeregular variationtail dependenceKendall's tauSpearman's rho

MSC classification

Primary: 60E05: Distributions
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 587 - 608
Copyright
Copyright © Applied Probability Trust 2002 

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