Statistical Inference Based On Several Stochastic Process Degradation Models With Random Effects

With the rapid development of technology and high competition in market,the qualify of industrial products are continuously improving.Consequently,it becomes challenging to collect sufficient failure data for the evaluation of product reliability within a short time period.In many applications,some quality characteristics of the industrial products degrade over time,and a product fails when the amount of degra-dation reaches a critical level.Therefore,degradation data are widely used to assess the product reliability.A large amount of existing literature mainly rely on large sam-ple approximations for degradation model inference.Although large sample approx-imations tend to work well when a sufficient number of product units are available,their performance degenerates in the face of small sample size.In degradation test data analysis,the sample size is typically small because it is usually affordable to test many units.On the other hand,an accurate degradation model inference is important for reliability assessment and the subsequent system safety evaluation,and biases in reliability prediction could lead to serious safety issues.In this paper,by considering different degradation models,a statistical inference method that has a good perfor-mance under small sample size is proposed based on the generalized pivotal quantity.The main contents are summarized as follows:(1)We focus on the statistical inference based on the Wiener process with ran-dom drift parameter for degradation data.We propose an exact procedure to test whether there is population heterogeneity.An exact confidence interval(CI)proce-dure 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 simu-lation.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.(2)Based on the principle of degradation mechanism invariance,a Wiener degra-dation process with random drift parameter is used to model the data collected from the constant stress accelerated degradation test.Small sample statistical inference method for this model is proposed.Based on Fisher's method,a test statistic is pro-posed to test if there is unit-to-unit variability in the population.For reliability infer-ence,the quantities of interest are the quantile function,the reliability function,and the mean time to failure at the designed stress level.Because it is challenging to ob-tain exact confidence intervals(CIs)for these quantities,a regression type of model is used to construct pivotal quantities,and we develop generalized confidence intervals(GCIs)procedure for those quantities of interest.Generalized prediction interval for future degradation value at designed stress level is also discussed.A Monte Carlo simulation study is used to demonstrate the benefits of our procedures.Through sim-ulation comparison,it is found that the coverage probabilities of the proposed GCIs are better than that of the Wald CIs,and GCIs have good properties even when there is only a small number of test samples available.(3)We focus on the statistical inference based on the Gamma degradation model with random effects.A generalized p-value procedure is proposed to test whether there exist some heterogeneities among the degradation processes of different unit-s.Using the Cornish-Fisher expansion,an approximate confidence interval(CI)is obtained for the shape parameter.The generalized confidence intervals(GCIs)are derived for model parameters and commonly used reliability metrics(e.g.,the quan-tile,the reliability function of the lifetime)based on the generalized pivotal quantity method.Those inference procedures are also extended to the accelerated degradation case.The performances of the proposed GCIs are assessed by Monte Carlo simula-tions.In the simulation,we compared our methods with the Wald CIs and bootstrap-p CIs under moderate and large sample sizes.It is found that the performance of the GCI procedures are better than the Wald CIs and bootstrap-p CIs in terms of coverage probabilities.(4)We focus on the statistical inference using the Wiener degradation model with random initial degradation.For the fixed effect model,an exact confidence in-terval(CI)inference procedure for the diffusion parameter is obtained.Generalized confidence intervals(GCIs)are proposed for model parameters and some commonly used reliability metrics based on the generalized pivotal quantity method.In order to capture heterogeneities among the population,those inference procedures are also extended to account for the random effects.We also propose a generalized p-value procedure to test whether there exist certain heterogeneities among degradation pro-cesses of different units.The performances of the proposed GCIs are assessed by Monte Carlo simulations.In the simulation,we compare our methods with the Wald CIs and the bootstrap-p CIs.It is found that the GCIs outperform the other two ap-proaches in terms of coverage probability.