Remaining useful life estimation of units characterized by a bathtub-shaped degradation rate in the presence of random effects

This paper proposes a new time scaled Wiener process with random effects that has been specifically conceived to describe non monotone degradation phenomena characterized by a bathtub shaped degradation rate, here intended as derivative of the mean function. The main features of the proposed model are illustrated and discussed. The maximum likelihood estimation of its parameters is addressed. A failure threshold model is adopted to formulate and estimate the probability distribution function of the remaining useful life, which constitutes the core prognostic tool in condition based maintenance. As a motivating example, the proposed model is applied a set of real degradation data of MOS Field Effect Transistors, where, as a result of preliminary analyses, some of the process parameters are assumed to be random in order to describe the considerable heterogeneity observed between the degradation paths of different units. Obtained results demonstrate the utility and affordability of the proposed model.