Archimedean Copula Inference via Taylor-Mode AD

Abstract

No existing nested Archimedean copula tool handles all three of (a) arbitrary per-variable (right-)censoring in survival analysis, (b) arbitrary nesting trees, and (c) exact parameter gradients. Existing implementations handle only bivariate problems, low dimensional (i.e., $d \leq 10$) cases, two layers of nesting, or only hand-derived copula nestings. We present \textsc{acopula}, a JAX-native framework that, given any Archimedean generator -- classical or neural -- evaluates exact nested-copula likelihoods and parameter gradients under arbitrary censoring masks in polynomial time. The mechanism is polynomial powering of Taylor-mode automatic differentiation output, which replaces per-family hand-derived partial Bell polynomial tables with a single differentiable computation that any user-defined generator can drive. We conduct extensive simulations to verify the correctness of \textsc{acopula}. We then demonstrate (a) per-variable censoring on $85{,}229$ MIMIC-IV ICU admissions in high dimensions with $d{=}53$, fit by both classical Archimedean families and nested neural Archimedean copulas; (b) an 11-sector hierarchical model on S\&P~500 daily returns at $d{=}98$; (c) family-agnostic censored MLE across ten families, five of them with no prior implementation, on a retinopathy study; and (d) a ${\sim}650\times$ per-density speedup over R's \texttt{nacLL} at $d{=}35$, scaling quadratically to $d{=}8{,}000$.