Kernel Stein discrepancies (KSDs) have become a principal tool for goodness-of-fit testing, but standard KSDs are often insensitive to higher-order dependency structures, such as tail dependence, which are critical in many scientific and financial domains. We address this gap by introducing the Copula-Stein Discrepancy (CSD), a novel class of discrepancies tailored to the geometry of statistical dependence. By defining a Stein operator directly on the copula density, CSD leverages the generative structure of dependence, rather than relying on the joint density's score function. For the broad class of Archimedean copulas, this approach yields a closed-form Stein kernel derived from the scalar generator function. We provide a comprehensive theoretical analysis, proving that CSD (i) metrizes weak convergence of copula distributions, ensuring it detects any mismatch in dependence; (ii) has an empirical estimator that converges at the minimax optimal rate of $O_P(n^{-1/2})$; and (iii) is provably sensitive to differences in tail dependence coefficients. The framework is extended to general non-Archimedean copulas, including elliptical and vine copulas. Computationally, the exact CSD kernel evaluation scales linearly in dimension, while a novel random feature approximation reduces the $n$-dependence from quadratic $O(n^2)$ to near-linear $\tilde{O}(n)$, making CSD a practical and theoretically principled tool for dependence-aware inference.

翻译：核 Stein 差异已成为拟合优度检验的主要工具，但标准核 Stein 差异通常对高阶依赖结构（如尾部依赖）不敏感，而这些结构在许多科学与金融领域中至关重要。为填补这一空白，我们提出了 Copula-Stein 差异，这是一类针对统计依赖几何特性定制的新型差异度量。通过直接在 Copula 密度上定义 Stein 算子，CSD 利用了依赖的生成结构，而非依赖于联合密度的得分函数。对于广泛的阿基米德 Copula 类，该方法从标量生成器函数导出了封闭形式的 Stein 核。我们提供了全面的理论分析，证明 CSD：（i）能够度量 Copula 分布的弱收敛，确保其能检测依赖关系中的任何失配；（ii）其经验估计量以极小极大最优速率 $O_P(n^{-1/2})$ 收敛；（iii）对尾部依赖系数的差异具有可证明的敏感性。该框架可扩展至一般非阿基米德 Copula，包括椭圆 Copula 与藤 Copula。在计算方面，精确的 CSD 核评估在维度上呈线性扩展，而一种新颖的随机特征近似方法将 $n$ 依赖复杂度从二次 $O(n^2)$ 降低至近线性 $\tilde{O}(n)$，使得 CSD 成为一种实用且理论完备的依赖感知推断工具。