Copula-Stein Discrepancy: A Generator-Based Stein Operator for Archimedean Dependence

Kernel Stein discrepancies (KSDs) are widely used for goodness-of-fit testing, but standard KSDs can be insensitive to higher-order dependence features such as tail dependence. We introduce the Copula-Stein Discrepancy (CSD), which defines a Stein operator directly on the copula density to target dependence geometry rather than the joint score. For Archimedean copulas, CSD admits a closed-form Stein kernel derived from the scalar generator. We prove that CSD metrizes weak convergence of copula distributions, admits an empirical estimator with minimax-optimal rate $O_P(n^{-1/2})$, and is sensitive to differences in tail dependence coefficients. We further extend the framework beyond Archimedean families to general copulas, including elliptical and vine constructions. Computationally, exact CSD kernel evaluation is linear in dimension, and a random-feature approximation reduces the quadratic $O(n^2)$ sample scaling to near-linear $\tilde{O}(n)$; experiments show near-nominal Type~I error, increasing power, and rapid concentration of the approximation toward the exact $\widehat{\mathrm{CSD}}_n^2$ as the number of features grows.