Intensity-Based Estimation of Extreme Value Models for Highly Censored Data via a Latent Poisson Process

Abstract

Extreme value theory provides a natural framework for modelling rare events and extrapolating tail probabilities, but its direct application to survival analysis is complicated by right censoring, heterogeneous follow-up, and the need for coherent inference beyond the range of the observed data. We propose a Bayesian survival model that integrates extreme value theory with a hazard-based formulation built upon the extended generalized Pareto distribution. Rather than separating the bulk and tail of the distribution through an explicit threshold, the proposed model represents the entire hazard function within a unified parametric framework, allowing moderate and extreme survival times to be analysed simultaneously while preserving the asymptotic tail behaviour implied by extreme value theory. Inference is performed through a two-stage procedure. We further develop bias-correction procedures based on latent representative locations and adaptive partition refinement, and establish theoretical results showing the asymptotic consistency of the proposed approximation under standard regularity conditions. Simulation studies demonstrate that the methodology provides reliable uncertainty quantification across a wide range of tail regimes, censoring proportions, and sample sizes, while remaining robust under moderate model misspecification. The practical performance of the approach is illustrated through an analysis of a heart failure cohort, where the proposed model enables inference on long-term survival probabilities and tail behaviour beyond the observed follow-up period. The resulting framework offers a computationally feasible and theoretically grounded alternative for modelling censored survival data when inference on extreme survival times is of primary scientific interest.