IGNIS: A Robust Neural Network Framework for Constrained Parameter Estimation in Archimedean Copulas

TMLR Paper5511 Authors

Abstract: Classical estimators, the cornerstones of statistical inference, face insurmountable challenges when applied to important emerging classes of Archimedean copulas. These models exhibit pathological properties, including numerically unstable densities, a restrictive lower bound on Kendall's tau, and vanishingly small likelihood gradients, rendering methods like Maximum Likelihood (MLE) and Method of Moments (MoM) inconsistent or computationally infeasible. We introduce \textbf{IGNIS}, a unified neural estimation framework that sidesteps these barriers by learning a direct, robust mapping from data-driven dependency measures to the underlying copula parameter $\theta$. IGNIS utilizes a multi-input architecture and a theory-guided output layer ($\mathrm{softplus}(z) + 1$) to automatically enforce the domain constraint $\hat{\theta} \ge 1$. Trained and validated on four families (Gumbel, Joe, and the numerically challenging A1/A2), IGNIS delivers accurate and stable estimates for real-world financial and health datasets, demonstrating its necessity for reliable inference in modern, complex dependence models where traditional methods fail.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission:

During this revision, we made several important corrections and improvements. We sincerely apologise for any inconvenience. We also thank the reviewers for their insightful and constructive feedback, which has significantly improved the manuscript. We have addressed all comments, and the major revisions are summarized below. A detailed, point-by-point response to all reviewer comments is also provided in a separate PDF document included with this revised submission in the Supplementary Material Zip File along with an updated log-likelihood comparison code.

Correction of Kendall's $\tau$ for A1 : We discovered and fixed an algebraic error in the derivation of Kendall's $\tau$ for the A1 copula. The corrected formula (new Eq.\ 6) shows that $\tau_{A1}(\theta)$ is strictly increasing on $[1,\infty)$, as proven in Appendix A.3. As a consequence, interestingly both A1 and A2 now share the same lower bound $\tau_{\min} = 8\ln 2 - 5 \approx 0.54518$.

Removal of incorrect non-monotonicity claims: We removed all prior statements and simulations implying non-monotonicity of A1. Section 5.1 was updated accordingly.

Log-likelihood comparison updated: We re-ran the log-likelihood experiments with improved numerical stabilization and placed it in the main paper. The new Table 4 now includes \textbf{both A1 and A2} families. Results show that IGNIS achieves virtually identical out-of-sample log-likelihood to Method of Moments (MoM) where MoM applies, while extending estimation to cases where MoM cannot be defined.

Identifiability clarification: Appendix B was updated to reflect the corrected identifiability proof for A1, alongside the existing proof for A2.

All changes are marked in red in the manuscript.

We believe these revisions fully address the reviewers' comments and have resulted in a much stronger and clearer manuscript.

Assigned Action Editor: ~Vincent_Fortuin1

Submission Number: 5511