Title: On a class of measures of concordance for bivariate copulas
Authors: Sebastian Fuchs - University of Salzburg (Austria) [presenting]
Abstract: The purpose is to study a wide class of measures of concordance for bivariate copulas in which each element $kappa_A$ is generated by a fixed but arbitrary copula $A$. This class contains Spearman's rho, which is induced by the independence copula, and Gini's gamma, which is induced by the comonotonicity copula. Our approach sheds some new light on Spearman's rho and Gini's gamma and allows for the construction of other meaningful measures of concordance focussing on different facets of dependence between two random variables. In particular, we introduce a measure of concordance that is induced by the countermonotonicity copula. For all measures of concordance in this class, we propose a general construction of a sample version which is based on the empirical copula, and we show that these estimators are asymptotically normally distributed. For Spearman's rho and Gini's gamma it turns out that our sample versions coincide with the usual ones. Moreover, the pointwise order on copulas induces an order relation on this class of measures of concordance. It turns out that, for every copulas $C$ which is left tail decreasing and right tail increasing, the values $\kappa_A(C)$ are decreasing when $A$ is increasing and are bounded below by the corresponding values of Kendall's tau.