Advances in Applied Probability

Extremal behavior of Archimedean copulas

Martin Larsson and Johanna Nešlehová

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Abstract

We show how the extremal behavior of d-variate Archimedean copulas can be deduced from their stochastic representation as the survival dependence structure of an l1-symmetric distribution (see McNeil and Nešlehová (2009)). We show that the extremal behavior of the radial part of the representation is determined by its Williamson d-transform. This leads in turn to simple proofs and extensions of recent results characterizing the domain of attraction of Archimedean copulas, their upper and lower tail-dependence indices, as well as their associated threshold copulas. We outline some of the practical implications of their results for the construction of Archimedean models with specific tail behavior and give counterexamples of Archimedean copulas whose coefficient of lower tail dependence does not exist.

Article information

Source
Adv. in Appl. Probab. Volume 43, Number 1 (2011), 195-216.

Dates
First available in Project Euclid: 15 March 2011

Permanent link to this document
http://projecteuclid.org/euclid.aap/1300198519

Digital Object Identifier
doi:10.1239/aap/1300198519

Zentralblatt MATH identifier
05883120

Mathematical Reviews number (MathSciNet)
MR2761154

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

Keywords
Archimedean copula domain of attraction extreme value copula ell_1-norm symmetric distribution regular variation simplex distribution tail dependence threshold copula Williamson transform

Citation

Larsson, Martin; Nešlehová, Johanna. Extremal behavior of Archimedean copulas. Adv. in Appl. Probab. 43 (2011), no. 1, 195--216. doi:10.1239/aap/1300198519. http://projecteuclid.org/euclid.aap/1300198519.


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