Measures of association for approximating copulas

TL;DR

This paper derives closed-form formulas for various association measures of copulas, including Chatterjee's xi, and demonstrates how these can approximate the true dependence between variables with increasing accuracy as the grid size grows.

Contribution

It provides the first closed-form expressions for multiple copula association measures, notably including Chatterjee's xi, and analyzes their convergence properties.

Findings

Closed-form formulas for association measures of copulas.

Chatterjee's xi can be approximated by checkerboard copulas.

Approximation accuracy improves as grid size increases.

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 xi (also known as Chatterjee's rank correlation), which quantifies the dependence between two random variables. Given any bivariate copula $C$, we show that the closed-form formula for Chatterjee's xi of an approximating checkerboard copula serves as a lower bound that converges to the true value of $\xi(C)$ as one lets the grid size $n\rightarrow\infty$.