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.

翻译：核 Stein 差异（KSDs）被广泛用于拟合优度检验，但标准 KSDs 对高阶相依特征（如尾部相依）可能不敏感。我们引入了 Copula-Stein 差异（CSD），它直接在 Copula 密度上定义了一个 Stein 算子，以针对相依几何结构而非联合得分。对于阿基米德 Copula，CSD 允许一个从标量生成元导出的闭式 Stein 核。我们证明了 CSD 可度量 Copula 分布的弱收敛，允许一个具有极小极大最优速率 $O_P(n^{-1/2})$ 的经验估计量，并且对尾部相依系数的差异敏感。我们进一步将该框架扩展到阿基米德族之外，推广至一般 Copula，包括椭圆 Copula 和藤 Copula 结构。在计算上，精确的 CSD 核求值在维度上是线性的，而一种随机特征近似方法将二次的 $O(n^2)$ 样本复杂度降低至近线性的 $\tilde{O}(n)$；实验表明，随着特征数量的增加，该近似方法具有接近名义水平的 I 类错误、渐增的检验功效，并能快速收敛到精确的 $\widehat{\mathrm{CSD}}_n^2$。