Distorted Copulas: Constructions and Tail Dependence

Abstract

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1], is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ which ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behaviour in the tails.

On tail dependence coefficients of transformed multivariate Archimedean copulas

“…In particular the relationship between the regular variation of the transformation T and of the transformed tails is investigated. These results extend some bivariate results of Durante et al (2010) using the definition of upper and lower multivariate tail dependence coefficients previously proposed by De Luca and Rivieccio (2012). The first part of the paper exploits previous works of Charpentier and Segers (2007), and extends some bivariate results of Juri and Wüthrich (2003).…”

“…We consider the class of transformations previously proposed for instance in Durante et al (2010), Di Bernardino and Rullière (2013a), Di Bernardino and Rullière (2013b). In particular the relationship between the regular variation of the transformation T and of the transformed tails is investigated.…”

“…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.…”

Copula Theory: An Introduction

“…Such a transformation has originated from the study of distorted probability distribution functions (especially, power distortions), and has been considered by several authors like [3,18,42,52,54,73,85,112,113,150,146]. In reliability the-ory, this kind of transformation is used in order to introduce the so-called bivariate ageing function that are used for the definition of bivariate notion of ageing (see [8,9,53,148]).…”

Aging Functions and Multivariate Notions of Nbu and Ifr

“…It is obtained by transforming the copula K F by means of the bijection G • (− log) of I. Transformations of this type are known as distortions of copulas (see [10,14,21] and the references therein).…”