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

Technical Summary: Bivariate Inverse Gaussian Degradation Processes with Shared Random Effects

Problem Statement
The paper addresses the modeling of bivariate degradation data involving two performance characteristics (PCs) of engineered systems. While univariate degradation is frequently modeled using the Inverse Gaussian (IG) process, bivariate analysis presents the challenge of capturing the inherent dependence between the two PCs. Existing approaches typically rely on copula-based methods or frailty-based models. A significant limitation in current frailty-based literature is the assumption of a specific distribution for the shared frailty (e.g., normal, gamma, or lognormal) without empirical justification, as the frailty is an unobserved latent variable. Selecting an inappropriate frailty distribution can lead to sub-optimal model fitting and inaccurate reliability estimates.

Methodology
The authors propose a flexible IG–GG framework for modeling bivariate degradation data. The methodology consists of the following components:

1. Marginal Modeling: The degradation processes for the two PCs are modeled individually using Inverse Gaussian (IG) processes. Conditional on a shared random effect (frailty), the two processes are assumed to be independent.
2. Shared Frailty Distribution: Instead of fixing the frailty distribution to a specific form, the shared frailty $\lambda$ is assumed to follow the Generalized Gamma (GG) family of distributions. The GG family is a parametric class that includes the exponential, gamma, Weibull, and lognormal distributions as special cases (depending on the shape parameters $q$ and $\sigma$) and the lognormal as a limiting case.
3. Model Structure: The hierarchical model is defined as:
   • $X_1(t)|\lambda \sim IG(\lambda\mu_1\Lambda(t; b_1), \eta_1(\Lambda(t; b_1))^2)$
   • $X_2(t)|\lambda \sim IG(\lambda\mu_2\Lambda(t; b_2), \eta_2(\Lambda(t; b_2))^2)$
   • $\lambda \sim GG(q, \sigma, \alpha)$
     Here, $\Lambda(t; b_k)$ is a monotonic time transformation function.
4. Parameter Estimation:
   • Likelihood Construction: The marginal likelihood function is derived by integrating the conditional likelihood (given $\lambda$) over the GG distribution of the frailty. Since this integral lacks a closed-form expression, the authors employ numerical integration (quadrature methods via R's integrate function) and Monte Carlo sampling techniques to approximate the likelihood.
   • Optimization: Maximum Likelihood Estimates (MLEs) are obtained by numerically optimizing the marginal likelihood function. To ensure parameter identifiability, the mean of the shared frailty distribution is constrained to 1.
   • Model Selection: The framework allows for fitting all special members of the GG family (Exponential, Gamma, Weibull, Lognormal) alongside the general GG distribution. The most appropriate model is selected based on information criteria such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).
5. System Reliability: The paper derives an approximation for the system reliability function (the probability that both components survive beyond specific thresholds) by applying the bivariate central limit theorem to the discrete first-passage times and converting the result to a continuous form.

Key Results

• Simulation Studies: Extensive Monte Carlo simulations were conducted with varying sample sizes ($n=50, 100$) and measurement times ($m=20, 40$). The results demonstrated that the proposed estimation methods yield unbiased estimates with reasonable Root Mean Squared Errors (RMSE). The study confirmed that when the true underlying frailty distribution is a specific member of the GG family (e.g., Gamma or Weibull), the general GG model successfully identifies it or provides a fit comparable to the specific model, while avoiding the risk of misspecification.
• Case Study (Fatigue Cracks): The model was applied to a real-world dataset involving fatigue crack growth in two cracks (Crack A and Crack B) across 10 units. The IG–GG model was compared against several copula-based models (Gaussian, Frank, Gumbel, Clayton, FGM) and specific frailty-based models (IG with Exponential, Gamma, Weibull, and Lognormal frailties).
  • The IG–GG model achieved the highest log-likelihood and the lowest AIC and BIC values (-805.320 and -802.899, respectively), outperforming all copula-based and specific frailty-based alternatives.
  • The results indicated that the Generalized Gamma distribution provided a superior fit compared to assuming a fixed distribution like Gamma or Lognormal.

Significance and Claims
The paper claims that the primary significance of the proposed IG–GG framework lies in its flexibility. By utilizing the Generalized Gamma family, the model avoids the arbitrary and often unjustifiable assumption of a specific frailty distribution. This flexibility allows the data to dictate the most suitable distributional form for the shared random effect, leading to improved model fitting and more accurate representation of the dependence structure between degrading components.

The authors assert that this approach is particularly valuable because the frailty is unobserved, making statistical hypothesis testing for distribution selection impossible. The IG–GG framework circumvents this by fitting the entire family and selecting the best fit via information criteria. The paper concludes that this method provides a superior fit for bivariate degradation data compared to existing copula-based and fixed-frailty models, as demonstrated by the fatigue crack application, and offers a viable path for estimating system reliability in such contexts.