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Measures of association for approximating copulas

Marcus Rockel

TL;DR

This work delivers closed-form expressions for multiple association measures, notably Chatterjee's xi, across copula-approximation families such as Bernstein, shuffle--of--min, and checkerboard-type copulas. It establishes that checkerboard copulas yield a computable lower bound $\xi(C^{\Delta}_{\Pi})$ to the true $\xi(C)$ and that this bound converges as the grid refines ($n\to\infty$). The authors derive explicit formulas for $\rho_S$, $\tau$, $\xi$, and tail dependence for each family, and provide a theory-supported estimator framework that uses checkerboard discretization to approximate $\xi$ from data with provable convergence rates. A practical estimator, with guidance on the grid size parameter $\kappa$, demonstrates favorable finite-sample performance and computational efficiency relative to standard estimators. Overall, the paper offers analytic tractability and scalable estimation for dependence measures in copula-based approximations, with direct implications for model fitting and variable selection in dependence-driven applications.

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 $ξ(C)$ as one lets the grid size $n\rightarrow\infty$.

Measures of association for approximating copulas

TL;DR

This work delivers closed-form expressions for multiple association measures, notably Chatterjee's xi, across copula-approximation families such as Bernstein, shuffle--of--min, and checkerboard-type copulas. It establishes that checkerboard copulas yield a computable lower bound to the true and that this bound converges as the grid refines (). The authors derive explicit formulas for , , , and tail dependence for each family, and provide a theory-supported estimator framework that uses checkerboard discretization to approximate from data with provable convergence rates. A practical estimator, with guidance on the grid size parameter , demonstrates favorable finite-sample performance and computational efficiency relative to standard estimators. Overall, the paper offers analytic tractability and scalable estimation for dependence measures in copula-based approximations, with direct implications for model fitting and variable selection in dependence-driven applications.

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 , 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 as one lets the grid size .
Paper Structure (15 sections, 7 theorems, 102 equations, 3 figures)

This paper contains 15 sections, 7 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Copulas
  4. Bernstein copulas
  5. Shuffle--of--min copulas
  6. Checkerboard, check--min and check--w copulas
  7. Measures of association
  8. Explicit measures of association for approximating copulas
  9. Explicit measures of association for Bernstein copulas
  10. Explicit measures of association for shuffle--of--min copulas
  11. Explicit measures of association for checkerboard-type copulas
  12. Checkerboard estimates for Chatterjee's xi
  13. Checkerboard bound for Chatterjee's xi
  14. Checkerboard estimator for Chatterjee's xi
  15. Appendix

Key Result

Proposition 3.1

Let $C = C^D_B$ be the Bernstein copula associated with the cumulated $m\times n$-checkerboard matrix $D$. Then: Furthermore, the tail dependence coefficients are given by $\lambda_L(C^D_B) = \lambda_U(C^D_B) = 0$.

Figures (3)

  • Figure 1: The estimator $\xi^{\kappa}_n$ for different values of $\kappa$. Each boxplot corresponds to an increasing sample size $n$. The estimates concentrate near the theoretical value of $\xi$ (red line), illustrating consistency.
  • Figure 2: Convergence of xi estimates to the true value as sample size increases. As suggested by Theorem \ref{['thm:xi_bounds']}, the checkerboard estimate $\underline{\xi_n^{\kappa}}$ (CheckPi) tends to underestimate the true value, while the check--min estimate $\mkern 1.5mu\overline{\mkern-1.5mu\xi_n^{\kappa}\mkern-1.5mu}\mkern 1.5mu$ (CheckMin) tends to overestimate it. $\xi^{\kappa}_n$ (CheckAvg) is the closest to the true value at smaller sample sizes in this setting.
  • Figure 3: Execution time scaling for different estimation methods with increasing sample size. Our implementation of $\xi^{\kappa}_n$ outperforms the implementations of $\xi_n$ in xicorpy approximately by a factor of three and the implementation in scipy by approximately 30 %.

Theorems & Definitions (16)

  • Proposition 3.1: Explicit measures of association for Bernstein copulas
  • Proposition 3.2: measures of association for a straight shuffle--of--min copula
  • Proposition 3.3: Explicit measures of association for checkerboard copulas
  • Corollary 3.4
  • Theorem 4.1: Checkerboard bound for $\xi$
  • proof
  • Example 4.2
  • Theorem 4.3: Convergence of $\xi^{\kappa}_n$
  • proof
  • Example 4.4
  • ...and 6 more