Abstract

In real life situations, for example, in case of concrete structures and mechanical devises, failure occurs due to accumulation of stress over time, and the consequent degradation causes the system to under-perform, thereby failure occurs completely, when the accumulated stress exceeds a critical threshold. The degradation path is known for some systems and in such cases useful information on product reliability can be obtained from these measurements. This work considers a degradation model having exponential degradation path with positive degradation rate, which follows Weibull distribution with known shape parameter β, and unknown scale parameter α. The Bayesian analysis is performed by considering noninformative (Quasi) and informative (Gamma) prior distributions for the scale parameter α. The Bayesian estimators of α and the system reliability are presented as well. The standard error for estimated α corresponding to both informative and noninformative priors are calculated using bootstrap method. It is observed that for larger bootstrap sample, the Bayes estimator of α, tends to have minimum standard error. It is also seen from the simulation study carried out by making use of Gibbs sampling technique that the Bayesian reliability of the system approaches the actual reliability for increasing sample size.

摘要

在现实生活中，例如，在混凝土结构和机械设备的情况下，由于应力随时间累积而发生故障，随之而来的退化导致系统性能不佳，从而在累积应力超过 a 时完全发生故障临界阈值。某些系统的退化路径是已知的，在这种情况下，可以从这些测量中获得有关产品可靠性的有用信息。这项工作考虑了具有正退化率的指数退化路径的退化模型，该退化模型服从具有已知形状参数β和未知尺度参数α的 Weibull 分布. 贝叶斯分析是通过考虑尺度参数α的非信息性（准）和信息性（伽玛）先验分布来执行的。还介绍了α和系统可靠性的贝叶斯估计量。使用 bootstrap 方法计算对应于信息性和非信息性先验的估计α的标准误差。据观察，对于较大的自举样本，α的贝叶斯估计量往往具有最小标准误差。从利用吉布斯抽样技术进行的仿真研究中也可以看出，随着样本量的增加，系统的贝叶斯可靠性接近于实际可靠性。