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

2025-10-28

AI Summary

Standard kernelized Stein discrepancy (KSD) is insensitive to higher-order statistical dependence structures—particularly tail dependence—limiting its applicability in financial and scientific modeling. Method: We propose the Copula-Stein Discrepancy (CSD), the first Stein method directly applied to copula densities. Leveraging Archimedean copula generator functions, we construct a novel Stein operator and closed-form kernel that explicitly capture geometric dependence structures. Contribution/Results: We prove CSD metrizes weak convergence of copula distributions and is provably sensitive to tail dependence. Equipped with random feature approximations, its empirical estimator achieves the optimal $O_P(n^{-1/2})$ convergence rate with near-linear computational complexity. Experiments demonstrate substantial improvements in dependence structure detection across diverse copula models, confirming both strong theoretical guarantees and practical scalability.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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 $ ilde{O}(n)$, making CSD a practical and theoretically principled tool for dependence-aware inference.

Problem

• Detects tail dependence in goodness-of-fit testing
• Defines Stein operator directly on copula density
• Provides scalable dependence-aware inference framework

Innovation

• Defines Stein operator on copula density directly
• Uses generator function for Archimedean copulas
• Provides linear scaling and near-linear approximation