A geometric investigation into the tail dependence of vine copulas

TL;DR

This paper investigates the extremal dependence properties of vine copulas, focusing on tail dependence coefficients and the shape of extreme sample clouds, using a geometric approach for various classes including extreme value and inverted extreme value copulas.

Contribution

It provides new theoretical insights into the tail dependence of vine copulas, especially for high-dimensional cases constructed from inverted extreme value copulas.

Findings

Analysis of tail dependence coefficients for different vine copula classes

Geometric characterization of extreme sample clouds in vine copulas

Extension of theory to high-dimensional vine copulas from inverted extreme value copulas

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.