Revisiting fatigue-life distributions of Birnbaum-Saunders type: Mathematical characterization, simulation study, and applications

The Birnbaum-Saunders (BS) distribution is a fundamental model in reliability engineering and risk analysis, relating failure time to the accumulation of incremental damage. This accumulation is traditionally assumed to follow normality, although such an assumption may fail under heavy-tailed processes with potentially unbounded variability We extend the BS distribution to a more general family driven by stable-type increments, designed to handle both moderate and extremely heavy tails. This extension is supported by a generalized convergence argument. We introduce novel estimators —based on tail and characteristic function methods— that effectively capture extreme tail behavior, even with moderate sample sizes. We illustrate the practical benefits of our approach by examining two real insurance-claim datasets (finite-variance versus infinite-variance). The first is the Allstate dataset, a large auto-claim dataset available on Kaggle, for which a stable BS model with finite variability emerges as sufficient. The second is a French motor third-party liability dataset whose exceptionally high individual claims justify an unbounded-variance model. These contrasting illustrations demonstrate how the extended BS framework can reveal whether infinite variability is warranted, guiding practitioners toward models that are both theoretically sound and empirically reliable for risk assessment. Overall, our findings broaden the applicability of the BS distribution to extreme-event scenarios, providing a flexible framework capable of distinguishing between finite- and infinite-variance behaviors in contemporary reliability and actuarial applications. • It extends the Birnbaum–Saunders model to be generated from stable-type damages. • It unifies finite-variance and infinite-variance regimes within one framework. • It introduces novel Hill and characteristic-function methods for parameter estimation. • It provides two illustration with two insurance claim datasets of finite and infinite variance. • It provides a flexible approach for risk modeling in reliability and actuarial contexts.