A New Class of Markovian Processes for Deteriorating Units With State Dependent Increments and Covariates

We present a new class of increasing, continuous Markovian degradation processes, called transformed gamma processes, where the distribution of the degradation increment in a future time interval depends both on the current age and the current degradation level. Unlike other increasing, age- and state-dependent processes in the literature, transformed gamma processes are mathematically and statistically easily tractable. Indeed, for such a new class of processes, the conditional distribution of the degradation growth over a generic time interval, given the current state of the unit, can be formulated in a closed form without resorting to time or state discretization or both. The main properties of transformed gamma processes, which can also incorporate time-invariant covariates, are discussed. The conditional distribution of the first passage time to a given threshold is also derived. Maximum likelihood estimators of the model parameters are developed, that can be used on the basis of very general datasets of degradation measures, and a simulation study is carried out to assess the statistical properties of the estimators. A formal test is also presented that can be used to check whether, within the proposed class of models, the observed degradation process actually depends on unit age or state or both. An application, based on a real set of degradation data, is used to illustrate the potentiality of the transformed gamma processes in an applicative framework.