Quantifying extremal dependence is a central problem of multivariate extreme value theory and its statistical applications. Two forms of extremal dependence are possible: extremes are either asymptotically independent or asymptotically dependent.
Theoretical examples of the two limiting behaviours are the bivariate Normal distribution, for asymptotic independence, and the bivariate t distribution, for asymptotic dependence (see, for example, Ledford and Tawn, 1996, and Demarta and McNeil, 2005). In this work, we focus on the tail dependence of skewed extensions of these basic models, namely the bivariate skew-Normal and skew-t distributions (Azzalini and Capitanio, 1999 and 2003).
In recent years, the skew-Normal and the skew-t distributions, in their univariate and multivariate versions, have received considerable attention both in theoretical studies, for their numerous stochastic properties, and in applied studies, for the additional flexibility that they provide for modelling phenomena that depart from symmetry. For a comprehensive review see Azzalini (2005). Since these families have become standard tools in many areas of applications, including selective sampling, stochastic frontier and financial studies, it is of interest to investigate the type of tail behaviour which is implied by their use.
As a practical example, whenever multivariate financial return data are modelled through the skew-Normal distribution or the skew-t distribution (Adcock, 2004; Walls, 2005), it is crucial to know whether the adopted model admits the occurrence of simultaneous large losses (or large gains), i.e. whether returns are treated as asymptotically dependent or asymptotically independent. In addition, if the two distributions demonstrate to cover a wide range of asymptotic behaviours, they may provide useful models for extremal dependence as alternatives to those presently available in the literature (Ledford and Tawn, 1997; Bortot et al., 2000; Heffernan and Tawn, 2004).
We will show that both the skew-Normal distribution and the skew-t distribution inherit the asymptotic behaviour of the generating distribution, the skew-Normal being asymptotically independent and the skew-t being asymptotically dependent. However, within their respective limiting class, each provides a wider range of extremal dependence strength than the generating family. Moreover, both the skew-Normal and the skew-t distributions are shown to allow different upper and lower tail dependence.
The structure of the paper is as follows. Section 2 gives an overview of tail dependence measures proposed in the literature for asymptotically dependent and independent distributions. In Sections 3 and 4 respectively, the asymptotic characteristics of the bivariate skew-Normal distribution and of the bivariate skew-t distribution are illustrated. Technical proofs of the stated results are postponed to the Appendix. Finally, Section 5 contains some concluding remarks and ideas for future work.
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