On tail dependence coefficients of transformed multivariate Archimedean copulas

Abstract

This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate tail dependence coefficients when the generator of the considered copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. This first part exploits previous works of Charpentier and Segers (2009) and extends some results of Juri and Wüthrich (2003) and De Luca and Rivieccio (2012). In the second part of the paper we analyse the impact in the upper and lower multivariate tail dependence coefficients of a large class of transformations of dependence structures. These results are based on the transformations exploited by Di Bernardino and Rullière (2013a) and Di Bernardino and Rullière (2013b). We extend some bivariate results of Durante et al. (2010) in a multivariate setting by calculating multivariate tail dependence coefficients for transformed copulas. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. In the third part, we show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. Furthermore, using results in Larsson and Nešlehová (2011), we detail the extreme behaviour of the transformed radial part of Archimedean copulas. At last, we explain possible applications with Markov chains with specific dependence structure.

Approximate Bayesian Inference in Semiparametric Copula Models

“…Multivariate extensions of tail dependence indices are not yet fully developed. An interesting proposal for Archimedean copulae is discussed in Di Bernardino and Rullière (2016); consider a random vector X = (X 1 , X 2 , · · · , X d ), more precisely its version lying in the copula space U = (U 1 , U 2 , · · · , U d ) and denote by I the set {1, 2, . .…”

“…As an alternative, Di Bernardino and Rullière (2016) propose to estimate the multivariate tail dependence indices through estimation of the copula generator; however, if we assume to have no information about the shape of the copula function, it is difficult to assess the estimation error in this way. On the other hand, our approach may be easily extended to this multivariate setting.…”

Multivariate Analysis of Joint Probability of Different Rainfall Frequencies Based on Copulas

“…The tail dependence analysis is critical to investigating the magnitude of dependence in the upper and lower tails of a bivariate distribution [32]. It also helps to identify the most suitable copula by emphasising the joint occurrence of extreme values [33].…”

Unit-Gamma Distortion of Bivariate Copulas

“…Fischer and Köck (2010) present a general construction scheme of d-variate copulas, which generalizes the Archimedean family, study admissible conditions on the distortion functions and derive the tail dependence coefficients for power and dual power distortions. Di Bernardino and Rullière (2016) consider transformations using conversion functions and examine the impact of the transformations on tail dependence coefficients; see also references on transformations of copulas therein. Genest et al (1998) provide four valuable rules that can be combined to construct classes of Archimedean generators from an initial generator φ, including the right-composition rule stating that if h :…”