Statistical Inference for a Virtual Age Reliability Model

Abstract

During the lifetime of a system, repairs may be performed when the system fails. It is most common to assume either perfect repair or minimal repair. However, a repair actually will sometimes be between minimal repair and perfect repair, which is called imperfect repair. The Kijima type I virtual age model can be used to model these types of repairable systems. This model contains a parameter which reflects the restoration level after each repair. This thesis considers statistical inference for the Kijima type I model, which deals with repairable systems that can be restored to the operating state through system replacement or repair after the system fails. We present Bayesian analysis for the Kijima type I virtual age model, including consideration of the system's overall time to failure if a given number of repairs is possible. We use both Bayesian analysis, which specifies a single prior distribution, and a robust Bayesian analysis approach. A set of prior distributions is used in robust Bayesian analysis in order to deal with uncertainty regarding prior knowledge of the Kijima type I model parameters in a flexible way and to enhance the objectivity of the analysis in an imprecise Bayesian framework by computing predictive posterior distribution bounds for the reliability function of the system. Finally, we discuss the use of the developed methods to decide about optimal replacement. Optimal replacement is the methodology of replacing a system component at the most advantageous or efficient moment to increase its performance and minimize overall expected costs. Two policies are introduced with cost functions based on time and number of failures to make a decision on optimal replacement time or optimal number of failures of the system under the Kijima type I model using the Weibull distribution. These policies illustrate how the Bayesian and robust Bayesian analysis can be used for inferences about the optimal replacement and the expected total cost.