Hybrid, Causal, and Stochastic Field Modeling of Censored Heavy-Tailed Insurance Losses

Abstract

Insurance claim severities are typically positive, heavy-tailed, and subject to
reporting thresholds or deductibles, leading to left-censored observations.
Standard parametric models such as the log-normal distribution are often inadequate in this setting, while purely machine learning approaches lack interpretability, formal inferential guarantees, and the ability to support counterfactual analysis. This paper proposes a unified framework for modeling censored insurance losses that combines robust parametric modeling, machine learning, and causal inference. We introduce a left-censored log-Student-\(t\) regression model that accommodates heavy tails and nests the log-normal specification as a limiting case. To capture nonlinear covariate effects beyond the parametric structure, we augment this baseline with a residual-based XGBoost correction, yielding a hybrid estimator that preserves interpretability while improving predictive accuracy. We establish consistency and asymptotic normality of the parametric estimator and prove consistency of the hybrid predictor. Beyond prediction, the framework integrates modern causal machine learning methods, including Double Machine Learning and Causal Forests, to estimate partial and heterogeneous effects and to conduct counterfactual analysis. We further demonstrate how causal estimates can be incorporated into risk measures, leading to causal adjustments of Value-at-Risk and extensions to extreme value analysis. Simulation studies and an empirical application to automobile insurance data show substantial gains over traditional approaches, particularly under moderate to severe censoring. At the end we add a stochastic approach for censored insurance loss surfaces.
Overall, the proposed methodology provides a statistically principled and decision-relevant approach to insurance loss modeling under censoring and heavy
tails