Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

10.5 Tail Dependence Parameters, Functions, and Tail Order Functions

We begin this section with a discussion on tail dependence coefficients, introducing this concept and how to interpret the properties of models with this feature. Then we move to relaxing this definition to nonasymptotic cases by considering the case of tail dependence functions and the related tail order functions. The concept of tail dependence parameters, tail dependence functions, or tail order functions each play a crucial role in both copula modeling as well as extreme value theory.

10.5.1 TAIL DEPENDENCE COEFFICIENTS

Tail dependence provides one approach to quantification of the dependence in extremes of a multivariate distribution. Traditionally this notion of dependence was considered from a pairwise construction due to tractability of expressions for the pairwise construction when applied to copula models. However, there is no reason to restrict this notion to just pairwise analysis and later we consider first the pairwise definition and then the generalized definition for d-variate random vectors.

The importance of thinking about tail dependence was succinctly summarized in the questions posed in Charpentier (2003) as detailed later:

1. If one considers data that are is taken from a multivariate distribution anywhere in its support, then through the measures of dependence just discussed and dependence concepts previously detailed in this chapter, it is possible to obtain all the overall dependence structure ...