Copula Discrepancy: Benchmarking Dependence Structure

Abstract

We study a simple statistic for benchmarking how well a sample preserves a known bivariate dependence structure. Given a target copula family (Clayton or Gumbel) and parameter $\theta_P$, the Copula Discrepancy (CD) compares the target Kendall's tau $\tau(\theta_P)$ with the Kendall's tau implied by a parameter $\hat{\theta}$ fitted to the sample within the target family, i.e., $|\tau(\theta_P)-\tau(\hat{\theta})|$. We develop a moment-based version, prove consistency, asymptotic normality, and robustness results under i.i.d.\ sampling, and use an MLE-based version empirically for greater power against tail-structure misspecification. Building on this, we define two information-theoretic copula summaries, a copula KL divergence (CKL) and a copula entropy gap (CED), and establish basic consistency and central limit results for their plug-in estimators. In controlled experiments, CD reliably separates on-target and off-target copulas with matched Kendall's $\tau$, provides a dependence-aware signal for tuning SGLD step sizes where Effective Sample Size favors overly aggressive (and biased) settings, and remains stably nonzero under deliberate tail-dependence mismatch where a naive $\tau$-based diagnostic fails; CKL and CED offer a complementary Shannon-style view that echoes these findings. Timing benchmarks show that both CD variants incur only millisecond-level overhead over the tested range and exhibit near-linear empirical scaling in sample size, providing a lightweight, dependence-focused complement to quadratic-cost omnibus discrepancies such as the Kernel Stein Discrepancy (KSD).