Extremes

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

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

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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References

  1. Balkema, A.A., de Haan, L.: Residual life time at great age. Ann. Probab. 2, 792–804 (1974)MATHGoogle Scholar
  2. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes. Wiley, Chichester (2004)MATHCrossRefGoogle Scholar
  3. Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events, 2nd ed., Birkhäuser, Basel (2004)Google Scholar
  4. Fougères, A.-L.: Multivariate extremes. In: Finkenstädt, B., Rootzén, H. (eds.) Extremes Values in Finance, Telecommunications, and the Environment, pp. 373–388. CRC, Boca Raton, Florida (2004)Google Scholar
  5. Galambos, J.: The Asymptotic Theory of Extreme Order Statistics, 2nd ed., Krieger, Malabar (1987)MATHGoogle Scholar
  6. Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997)MATHGoogle Scholar
  7. Nelsen, R.B.: An Introduction to Copulas, 2nd ed., Springer, Berlin Heidelberg New York (2006)MATHGoogle Scholar
  8. Pickands, J. III: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)MATHMathSciNetGoogle Scholar
  9. Pickands, J. III: Multivariate extreme value distributions. Bull. Internat. Statist. Inst. Proc. 43th Session ISI (Buenos Aires), 859–878 (1981)Google Scholar
  10. Reiss, R.-D., Thomas, M.: Statistical Analysis of Extreme Values, 2nd ed., Birkhäuser, Basel (2001)MATHGoogle Scholar
  11. Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, Berlin Heidelberg New York (1987)MATHGoogle Scholar
  12. Roberts, A.W., Varberg, D.E.: Convex Functions. Academic, New York (1973)MATHGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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