Modeling two-scale degradation with heterogeneity: A unified random-effects inverse Gaussian framework

Accurate modeling of product degradation is essential for reliable life prediction and maintenance planning. In many applications, degradation is jointly driven by age (calendar time) and usage (cumulative operation), and further complicated by unit-to-unit heterogeneity due to manufacturing or environmental variations. While existing models often focus on single-scale processes or assume homogeneous systems, few frameworks can flexibly accommodate both two scales and population-level variability. This article proposes a unified two-scale degradation model based on additive re-parameterized inverse Gaussian processes with random effects. The model captures monotonic degradation under the joint influence of age and usage, and introduces two forms of random effects to flexibly capture varying degrees of heterogeneity across units. It provides closed-form expressions for failure time distributions, enabling efficient reliability analysis. Two inference methods are developed: (1) a maximum likelihood estimator via an expectation-maximization algorithm with bootstrap intervals, and (2) a Bayesian approach using Hamiltonian Monte Carlo sampling. Simulation studies confirm the accuracy and robustness of the proposed estimators and highlight the risks of model misspecification in reliability assessment. A real-world case study on outdoor coating degradation further demonstrates the model’s practical applicability. An open-source R package is provided to support implementation, with additional materials available online.