A new Wiener process with 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 specially designed to allow the description of non-monotonic degradation phenomena with bathtub shaped degradation rate, here intended as derivative of the mean function. Two different parameterizations of the proposed Wiener process are suggested, and the main features of the process 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 to a set of real degradation data of metal-oxide-semiconductor 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 affordability and utility of the proposed model, especially for the estimation of unit-specific features performed on the basis of degradation data collected in the early phase of the life of a unit belonging to the considered heterogeneous population.