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

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 AA is a rescaled version of the volume of the area under the graph of AA, 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 AA.

Keywords

• asymmetry
• copula
• dependence measure
• exchangeability
• markov kernel
• 62h20 (primary), 60e05 (secondary)