Bivariate inverse Gaussian degradation processes with shared random effects and an application to fatigue cracks

2026-06-03

AI Summary

This study addresses the limitations of conventional approaches in modeling dependence between performance characteristics in bivariate degradation data, which often impose rigid and unverifiable assumptions on the distribution of shared frailty factors. To overcome this, the authors propose a bivariate degradation model based on inverse Gaussian processes coupled with a shared random effect following a generalized gamma distribution. The generalized gamma formulation flexibly encompasses several classical distributions, substantially enhancing modeling adaptability and interpretability. Integrated with an efficient parameter estimation algorithm and a comprehensive system reliability assessment framework, the proposed model demonstrates superior fitting accuracy and reliability estimation compared to existing frailty-based models and Copula methods, as validated on both simulated and real-world fatigue crack growth data.

Abstract

The inverse Gaussian (IG) process is a widely used model for univariate degradation data. For bivariate degradation data involving two performance characteristics (PCs), dependence is often introduced through an unobserved shared frailty factor combined with IG processes. Previous studies typically assume a specific frailty distribution, such as normal or gamma, although such choices are difficult to justify because the frailty is unobserved. This paper proposes a general IG GG framework for modeling bivariate degradation data with dependent PCs. Each degradation process is modeled using an IG process, while the shared frailty follows the generalized gamma (GG) family, which includes exponential, gamma, Weibull, and lognormal distributions as special cases. The proposed framework allows flexible selection of an appropriate frailty distribution within the GG family, leading to improved model fitting. Convenient parameter estimation procedures are developed and evaluated through simulation studies, demonstrating satisfactory performance. The proposed model is applied to fatigue crack data and compared with several existing frailty based and copula based models. Results show that the IG GG model provides a superior fit. System reliability estimation under the IG GG framework is also discussed.

Problem

• bivariate degradation
• inverse Gaussian process
• shared frailty
• generalized gamma distribution
• model fitting

Innovation

• inverse Gaussian process
• generalized gamma distribution
• shared frailty
• bivariate degradation modeling
• system reliability