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Extremes

, Volume 9, Issue 1, pp 63–68 | Cite as

A representation of bivariate extreme value distributions via norms on \( \mathbb{R}^{2} \)

  • Michael FalkEmail author
Article

Abstract

It is known that a bivariate extreme value distribution (EVD) \(G\) with reverse exponential margins can be represented as \(G(x,y)=\exp(-||(x,y)||)\), \(x,y\le 0\), where \(||\cdot||\) is a suitable norm on \(\mathbb{R}^2\). We prove in this paper the converse implication, i.e., given an arbitrary norm \(||\cdot||\) on \(\mathbb{R}^2\), \(G(x,y):=\exp(-||(x,y)||)\), \(x,y\le 0\), defines an EVD with reverse exponential margins, if and only if the norm satisfies for \(z\in[0,1]\) the condition \(\max(z,1-z)\le ||(z,1-z)||\le 1\). This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD \(G(x,y)=\exp(-||(x,y)||)\), \(x,y\le 0\), with the unit ball corresponding to the generating norm \(||\cdot||\), we obtain a characterization of the class of EVDs \(G\) in terms of compact and convex subsets of \(\mathbb{R}^2\).

Keywords

Bivariate extreme value distribution Pickands dependence function Norm Extreme value copula Convex set 

AMS 2000 Subject Classification

Primary—60G70 

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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