A Wiener process with generalized heterogeneity and volatility-rate dependence for degradation analysis

Unit-to-unit variability is commonly observed in the degradation of a population due to unobservable factors, such as variations in raw materials, usage patterns, and other influences. In the literature, this heterogeneity in degradation rates or volatility is often treated as a random effect. However, in practice, degradation rate and volatility may be interdependent. This study develops a Wiener degradation model in which degradation volatility is modeled as a function of the degradation rate. Additionally, a generalized inverse Gaussian distribution is employed to describe unit-specific heterogeneity, effectively capturing potential skewness and heavy-tailed characteristics. We derive a generalized closed-form expression for the lifetime distribution of the proposed model, incorporating different combinations of time-scale transformation functions. A statistical inference scheme based on the expectation-maximization algorithm, along with an interval estimation strategy, is presented. The effectiveness and applicability of the proposed model and corresponding algorithm are validated through comprehensive numerical simulations and two real-world datasets.