Measures of association for approximating copulas

This paper studies closed-form expressions for multiple association measures of copulas commonly used for approximation purposes, including Bernstein, shuffle–of–min, checkerboard and check–min copulas. In particular, closed-form expressions are provided for the recently popularized Chatterjee’s $\xi$, which quantifies the dependence between two random variables. Given an absolutely continuous bivariate copula $C$ with TP${}_2$ density and approximating $n\times n$-checkerboard copula $C_n$, we show that $\xi (C_n)\le \xi (C)$ with $\xi (C_n)\to \xi (C)$ as $n\to \mathrm{\infty }$.