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

Authors:	Michael Falk

Affiliation:	(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 , , where is a suitable norm on $mathbb{R}^2$ . We prove in this paper the converse implication, i.e., given an arbitrary norm on $mathbb{R}^2$ , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , 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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