Dependence measures for perturbations of copulas

Abstract

In this paper (which is a substantially extended version of a conference paper from AGOP 2015), we investigate the effects of specific class of perturbations of bivariate copulas on several measures of dependence (Spearman's rho, Blomqvist's beta, Gini's gamma, Kendall's tau), and tail dependence along both diagonal sections. It is demonstrated that the influence of the perturbation parameter on the values of the first three of the above coefficients of dependence is linear, while on the last one it is quadratic. Interesting numerical analyses for several important classes of Archimedean copulas are presented. It is also demonstrated that the considered perturbations do not change the coefficients of tail dependencies along the main diagonal but linearly reduce their values along the second diagonal. An interesting possible application for analyzing dependencies along the second diagonal of copulas represent insurance data, where censoring introduces a negative dependence between the investigated components of the claims. As a by-product, we present a new class of perturbations of copulas that linearly reduce the more popular coefficients of tail dependencies along the main diagonal, while preserving their values along the second diagonal. Subsequently using suitable elements of both above classes of perturbations, any original copula can be transformed to a resulting one, having coefficients of tail dependencies along both diagonals linearly reduced (with any couple of preselected linear proportions from [0,1]).