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

Agnideep Aich, Ashit Baran Aich

TL;DR

Copula-Stein Discrepancy (CSD) introduces a dependence focused goodness-of-fit metric by defining a Stein operator directly on the copula density. The framework yields a closed-form Copula-Stein kernel for Archimedean copulas and extends to general copulas, with rigorous guarantees that it metrizes weak copula convergence and achieves the minimax $O_P(n^{-1/2})$ rate while being sensitive to tail dependence. Computationally, CSD scales linearly in dimension with exact kernel evaluation and supports a random-feature approximation that lowers complexity to near-linear in $n$, enabling scalable dependence testing on large datasets. The approach provides a unified, principled tool for dependence-aware diagnostics across Archimedean, elliptical, vine and mixture copulas, with broad potential extensions to two-sample testing, parameter estimation, and causal inference.

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

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

TL;DR

Copula-Stein Discrepancy (CSD) introduces a dependence focused goodness-of-fit metric by defining a Stein operator directly on the copula density. The framework yields a closed-form Copula-Stein kernel for Archimedean copulas and extends to general copulas, with rigorous guarantees that it metrizes weak copula convergence and achieves the minimax rate while being sensitive to tail dependence. Computationally, CSD scales linearly in dimension with exact kernel evaluation and supports a random-feature approximation that lowers complexity to near-linear in , enabling scalable dependence testing on large datasets. The approach provides a unified, principled tool for dependence-aware diagnostics across Archimedean, elliptical, vine and mixture copulas, with broad potential extensions to two-sample testing, parameter estimation, and causal inference.

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 ; 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 -dependence from quadratic to near-linear , making CSD a practical and theoretically principled tool for dependence-aware inference.

Paper Structure

This paper contains 48 sections, 25 theorems, 107 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Kernel Stein Discrepancies
  4. Goodness-of-Fit Testing for Copulas
  5. Copula Modeling and Diagnostics
  6. Background: Copulas and Stein's Method
  7. Copulas and Dependence Modeling
  8. Kernel Stein Discrepancy
  9. Standing Assumptions and Notation
  10. What this section does.
  11. Distributions and Samples.
  12. Copula Framework.
  13. RKHS/Stein Framework.
  14. Standing Assumptions (used throughout).
  15. Strengthening used only for metrization.
  16. ...and 33 more sections

Key Result

Lemma 1

Under A1–A2, for any $g\in \mathcal{H}^d$ with each $g_j$ vanishing on the faces $\{u_j=0\}$ and $\{u_j=1\}$,

Theorems & Definitions (34)

  • Definition 4.1: Copula Stein discrepancy
  • Definition 5.1: Copula-Stein Operator
  • Lemma 1: Stein identity and boundary conditions
  • Proposition 5.1: Witness Function Representation
  • Theorem 1: Closed-Form CSD
  • Remark 5.1
  • Proposition 5.2: Explicit Kernel Formula
  • Proposition 5.3: Archimedean Score Function
  • Remark 5.2
  • Proposition 5.4: Computational Complexity
  • ...and 24 more