Kybernetika 47 no. 5, 732-739, 2011

Generalized Logistic Models and its orthant tail dependence

Helena Ferreira and Luisa Pereira

Abstract:

The Multivariate Extreme Value distributions have shown their usefulness in environmental studies, financial and insurance mathematics. The Logistic or Gumbel-Hougaard distribution is one of the oldest multivariate extreme value models and it has been extended to asymmetric models. In this paper we introduce generalized logistic multivariate distributions. Our tools are mixtures of copulas and stable mixing variables, extending approaches in Tawn \cite{tawn}, Joe and Hu \cite{joe+hu} and Fougères et al. \cite{fougeres+nolan+rootzen}. The parametric family of multivariate extreme value distributions considered presents a flexible dependence structure and we compute for it the multivariate tail dependence coefficients considered in Li \cite{li}.

Keywords:

tail dependence, logistic model, mixture, multivariate extreme value distribution

Classification:

60G70

paper.pdf

References:

  1. P. Capéraà, A. L. Fougères and C. Genest: Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 (2000), 30-49.   CrossRef
  2. C. M. Cuadras and J. Augé: A continuous general multivariate distribution and its properties. Comm. Statist. A - Theory Methods 10 (1981), 339-353.   CrossRef
  3. A.-L. Fougères, J. P. Nolan and H. Rootzén: Models for dependent extremes using scale mixtures. Scand. J. Statist. 36 (2009), 42-59.   CrossRef
  4. J. E. Heffernan, J. A. Tawn and Z. Zhang: Asymptotically (in)dependent multivariate maxima of moving maxima processes. Extremes 10 (2007), 57-82.   CrossRef
  5. H. Joe: Multivariate Models and Dependence Concepts. Chapman \& Hall, London 1997.   CrossRef
  6. H. Joe and T. Hu: Multivariate distributions from mixtures of max-infinitely divisible distributions. J. Multivariate Anal. 57 (1996), 240-265.   CrossRef
  7. H. Li: Orthant tail dependence of multivariate extreme value distributions. J. Multivariate Anal. 100 (2009), 243-256.   CrossRef
  8. A. W. Marshall and I. Olkin: Families of multivariate distributions. J. Amer. Statist. Assoc. 83 (1988), 834-841.   CrossRef
  9. A. J. McNeil, R. Frey and P. Embrechts: Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton 2005.   CrossRef
  10. P. M. Morillas: A method to obtain new copulas from a given one. Metrika 61 (2005), 169-184.   CrossRef
  11. R. B. Nelsen: An Introduction to Copulas. Springer, New York 1999.   CrossRef
  12. F. Schmid and R. Schmidt: Multivariate conditional versions of Spearman's rho and related measures of tail dependence. J. Multivariate Anal. 98 (2007), 1123-1140.   CrossRef
  13. R. L. Smith and I. Weissman: Characterization and Estimation of the Multivariate Extremal Index. Technical Report, Univ. North Carolina 1996.   CrossRef
  14. J. Tawn: Modelling multivariate extreme value distributions. Biometrika 77 (1990), 2, 245-253.   CrossRef