Measures of association for approximating copulas

Abstract

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 $\hat{I}¾$, 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 $\hat{I}¾(C_n)\le \hat{I}¾(C)$ with $\hat{I}¾(C_n)\to \hat{I}¾(C)$ as $n\to \infty$.

28 pages