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 transformed tail dependence coefficients when the generator of the considered transformed copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. These results are also valid for non-transformed Archimedean copulas. In the second part we deal with transformations presented by Di Bernardino and Rullière [20]. 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. Finally, we detail the extreme behaviour of the transformed radial part of Archimedean copulas (using results in Larsson and Nešlehová [52]) and we explain possible applications with transformed Markov chains.