Accurate reliability inference based on Wiener process with random effects for degradation data

The Wiener process is often used to fit degradation data in reliability modeling. Though there is an abundant literature covering the inference procedures for the Wiener model, their performance may not be satisfactory when the sample size is small, which is often the case in degradation data analysis. In this paper, we focus on the accurate reliability inference based on the Wiener process with random drift parameter for degradation data. We propose an exact procedure to test whether there is population heterogeneity. An exact confidence interval (CI) procedure for the diffusion parameter of the Wiener process is also obtained. Generalized confidence intervals (GCIs) are proposed for model parameters and some commonly used reliability metrics such as the quantile, the reliability function, etc. Furthermore, a generalized prediction interval (GPI) for the future degradation levels is obtained. The performance of the proposed GCIs and GPI is assessed by the Monte Carlo simulation. The simulation results show that the proposed interval procedures outperform existing method such as the Wald, the bootstrap-p and the likelihood-ratio-based CIs in terms of the coverage probability, and our proposed procedures have desirable properties even under small sample sizes. Finally, an example is used to illustrate the proposed procedures.