Modelling degradation data using the Gamma process and its generalizations

Many units degrade during their life and fail when their degradation level exceeds a prefixed threshold limit. Hence, once their degradation level is described via a proper stochastic process, it is possible to formulate the unit reliability in terms of the first passage time of the degradation level to the threshold limit. This approach, that allows to link the failure time of a degrading unit to its degradation level, offers several advantages. In particular, it permits the unit reliability to be estimated by using the degradation data even when failure data are not available, and enables one to predict the residual life in order to perform condition-based maintenance activities. Among the degradation processes proposed in the literature, the Gamma process is probably the widest applied in the reliability field. Its key success factors are flexibility and mathematical tractability, which allow dealing with many different kinds of degradation phenomena with a very limited computational burden. Relying on the latter feature, many authors have proposed generalizations of the gamma process, to use in specific practical settings. The aim of this paper is to survey some of these generalizations. In particular, we will focus on models that incorporate covariates, models that can account for the presence of random effects, models that assume the dependence of the degradation increment on the current degradation level (or state) of the units, and combinations thereof. All these models will be presented and examples of applications to real degradation data will be illustrated.