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

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, making MLE brittle and limiting MoM's applicability to datasets with sufficiently strong dependence (i.e., only when the empirical Kendall's $\tau$ exceeds the family's lower bound $\approx 0.545$). 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 $\hat{\theta} \geq 1$. 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. To our knowledge, IGNIS is the first \emph{standalone, general-purpose} neural estimator for Archimedean copulas (not a generative model or likelihood optimizer), delivering direct, constraint-aware $\hat{\theta}$ and readily extensible to additional families via retraining or minor output-layer adaptations.