Unobserved heterogeneity in stable imperfect repair models
Xingheng Liu, Jørn Vatn, Yann Dijoux, Håkon Toftaker
Abstract
This study investigates the effect of heterogeneity on the failures of repairable systems that undergo imperfect repairs, which are extensively used in reliability engineering. When considering a group of similar systems, the assumption that the repair processes are independent and identically distributed becomes questionable owing to the unobserved heterogeneity in these systems. The basic model we consider is the Kijima type II virtual age process with constant repair efficiency and a Weibull baseline distribution. We use the frailty models to study the heterogeneity between the systems and, in particular, the gamma-distributed frailty is investigated. We thus derive the asymptotic properties of the mixed repair process and corresponding likelihood estimates, and then evaluate the effects on the model parameter estimation process when heterogeneity is erroneously ignored. Furthermore, when the model is established correctly by accounting for the gamma distribution, we find that the maximum likelihood estimator is inconsistent and propose an alternative approach. Three case studies are presented to illustrate the benefits of taking account of unobserved heterogeneity in the estimation of the aging speed and reliability of assets and in scheduling preventive maintenance activities.