A link between Kendall’s τ, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support

Abstract

Abstract Working with shuffles, we establish a close link between Kendall’s τ \tau , the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well known that Spearman’s ρ \rho of a bivariate copula A A is a rescaled version of the volume of the area under the graph of A A , in this contribution we show that the other famous concordance measure, Kendall’s τ \tau , allows for a simple geometric interpretation as well – it is inextricably linked to the surface area of A A .