Maximum likelihood estimation in a bivariate Wiener degradation model with imperfect maintenance actions

Abstract This paper presents a statistical inference study for a bivariate degradation model incorporating imperfect maintenance. The system is subject to maintenance actions that only partially restore the degradation levels of both components. The underlying degradation process is modelled as a bivariate Wiener process, while imperfect maintenance is represented through an Arithmetic Reduction of Degradation (ARD$_{\infty }$) model. The system undergoes regular inspections, during which degradation levels of both components are measured simultaneously. The study examines several distinct observation schemes, which allow for measuring degradation levels both between maintenance actions and immediately before or after maintenance events. The maximum likelihood estimators of the model parameters are derived for the most general observation scheme. The cases in which degradation levels are measured just before or just after maintenance are treated as special cases of the general scheme. The accuracy and performance of the estimators are assessed through an extensive simulation study. The results show that the estimator of maintenance effectiveness performs very well, the estimators of the drift parameters are satisfactory, whereas the estimators of the diffusion parameters may be biased when no measurements are taken at maintenance times. The best estimates are obtained when measurements are taken just after maintenances. These results provide practical guidance: when feasible, measuring the degradation immediately after maintenance substantially improves parameter estimation.