Abstract
The strong and the weak tail dependence coefficients are measures that quantify the probability of conjoint extreme events of two random variables. Whereas formulas for both tail dependence coefficients exist for the Gaussian and Student t distribution, only the strong tail dependence coefficient is known for their super-model, the elliptical generalized hyperbolic distribution, which is extremely popular in finance (see Schmidt 2003). In this work we derive a simple expression for the corresponding weak tail dependence coefficient using the mixture representation of the elliptical generalized hyperbolic distribution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun I.A.: Handbook of Mathematical Functions, pp. 355–494. Dover, New York (1965)
Barndorff-Nielsen, O.: Exponential decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. Ser. A 353, 401–419 (1977)
Blæsild, P.: The two-dimensional hyperbolic distribution and some related distributions, with an application to Johannsen’s bean data. Biometrica 68, 251–263 (1981)
Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999)
Eberlein, E., Keller, U.: Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995)
Eberlein, E., von Hammerstein, E.A.: Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: Dalang, R.C., Dozzi, M., Russo F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability 58. Birkhäuser, Basel (2004)
Heffernan, J.E.: Directory of coefficients of tail dependence. Extremes 3(3), 279–290 (2000)
Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997)
Jørgensen, B.: Statistical properties of the generalized inverse Gaussian distribution. In: Lecture Notes in Statistics 9. Springer, New York (1982)
Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen. Teubner, Leipzig (1909)
Lawless, J.F.: Statistical Models and Methods for Lifetime Data. Wiley, New York (1982)
Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187 (1996)
Manner, H., Segers, J.: Tails for Correlation Mixtures of Elliptical Copulas. Technial Report, Institut de Statistique, Université Catholique de Louvain (2009)
McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton (2005)
Nelsen, R.: An Introduction to Copulas. Springer, New York (1999)
Reiss, R.-D.: Approximate Distributions of Order Statistics: With Applications to Non-Parametric Statistics. Springer, New York (1989)
Schmidt, R.: Tail dependence for elliptically contoured distributions. Math. Methods Oper. Res. 55, 301–327 (2002)
Schmidt, R.: Dependencies of Extreme Events in Finance. Dissertation, University of Ulm, Ulm (2003)
Sibuya, M.: Bivariate extreme statistics. Ann. Inst. Stat. Math. 11, 195–210 (1960)
Sklar, A.: Fonctions Deépartion à n Dimensions et Leurs Marges. Publications de l’Institut de Statistique de l’Université de Paris. l’Université de Paris, Paris (1959)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schlueter, S., Fischer, M. The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution. Extremes 15, 159–174 (2012). https://doi.org/10.1007/s10687-011-0132-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-011-0132-z