提出新型检验方法，精准捕捉变量尾部依赖关系。
Copula-Stein Discrepancy: A Generator-Based Stein Operator for Archimedean Dependence
- 在拷贝密度上定义斯坦因算子，直接针对依赖结构建模
- 对阿基米德类拷贝可闭式求解，采样误差率最优 $O_P(n^{-1/2})$
- 适用于椭球与藤结构等复杂依赖，计算高效可扩展
核斯坦因散度（KSD）广泛用于拟合优度检验，但对尾部依赖等高阶依赖特征敏感性不足。本文提出拷贝-斯坦因散度（CSD），直接在拷贝密度上定义斯坦因算子，以捕捉依赖几何结构而非联合得分。对于阿基米德拷贝，CSD 可基于标量生成器导出闭式斯坦因核。我们证明 CSD 能度量拷贝分布的弱收敛性，具有最小最大最优率 $O_P(n^{-1/2})$ 的经验估计器，并能敏感检测尾部依赖系数差异。进一步将框架拓展至椭球与藤结构等一般拷贝。计算上，精确 CSD 核评估随维度线性增长，随机特征近似将样本复杂度从二次 $O(n^2)$ 降至近线性 $ ilde{O}(n)$；实验表明其接近名义第一类错误率，检验功效随特征数增加而提升，且近似值快速收敛至精确 $\ ext{CSD}_n^2$。
原文摘要 · Abstract (English)
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.
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