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

Authors:	Michael Falk

Institution:	(1) Institute of Mathematics, University of Würzburg, D-97074 Würzburg, Germany

Abstract:	It is known that a bivariate extreme value distribution (EVD) 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 $z\in0,1]$ 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 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
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