Abstract
Vine copulas are a type of multivariate dependence model, composed of a collection of bivariate copulas that are combined according to a specific underlying graphical structure. Their flexibility and practicality in moderate and high dimensions have contributed to the popularity of vine copulas, but relatively little attention has been paid to their extremal properties. To address this issue, we present results on the tail dependence properties of some of the most widely studied vine copula classes. We focus our study on the coefficient of tail dependence and the asymptotic shape of the sample cloud, which we calculate using the geometric approach of Nolde (2014). We offer new insights by presenting results for trivariate vine copulas constructed from asymptotically dependent and asymptotically independent bivariate copulas, focusing on bivariate extreme value and inverted extreme value copulas, with additional detail provided for logistic and inverted logistic examples. We also present new theory for a class of higher dimensional vine copulas, constructed from bivariate inverted extreme value copulas.
Highlights
In multivariate extreme value analysis, the tail dependence properties of variables are an important consideration for model selection
For a copula based on the inverted bivariate extreme value distribution with underlying measure H placing all its mass at a finite number of atoms, the η{1,2} value follows from [23], with η{1,2} < 1 (unless H({1/2}) = 1), and the asymptotic shape of the sample cloud following from results in [20]
In Section A of the Supplementary Material, we show that the gauge function of a trivariate vine copula with three inverted extreme value components is g(x) = x2 + V{13|2}
Summary
In multivariate extreme value analysis, the tail dependence properties of variables are an important consideration for model selection. Since the tail dependence features of the Gaussian model are well-studied in the literature, we focus on cases where the pair copulas are from extreme value or inverted extreme value classes of distributions [24, 28]. These classes are widely studied in the extreme value literature; while they are not in themselves parametric distributions, they do include a range of well-known parametric examples [5, 10, 13]. Bivariate extreme value distributions exhibit asymptotic dependence, while their inverted counterparts exhibit asymptotic independence Studying these two classes is sufficient to reveal a rich variety of structures within the vine copula framework.
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