Tail dependence functions and vine copulas

Tail dependence and conditional tail dependence functions describe, respectively, the tail probabilities and conditional tail probabilities Of a Copula at various relative scales. The properties as well as the interplay of these two functions are established based upon their homogeneous structures. The extremal dependence of a, copula, as described by its extreme Value copulas, is shown to be completely determined by its tail dependence functions. For a vine Copula built from a set of bivariate copulas, its tail dependence function can be expressed recursively by the tail dependence and conditional tail dependence functions of lower-dimensional margins. The effect of tail dependence of bivariate linking copulas on that of a vine copula is also investigated. (C) 2009 Elsevier Inc. All rights reserved.