A new state-dependent degradation process and related model misidentification problems

The Gamma process with random scale parameter is often used to describe independent increments degradation phenomena in the presence of random effects. In this paper, a new Markovian degradation process, the Transformed Beta process, is proposed, where the degradation growth depends on both the current age and degradation level. It is also shown that the Transformed Beta process shares the same stochastic properties of the Gamma process whose scale parameter is Gamma-distributed, and that the likelihood functions relative to a given set of degradation data under these two models coincide. Due to this circumstance, it is not possible to choose between these two models on the basis of degradation data alone, even though they rely on quite different assumptions. On the other hand, although these models are stochastically equivalent, there are situations in which selecting the wrong model can produce real practical consequences. In fact, it is shown that, if the considered degrading unit is subjected to a so-called “virtual age imperfect maintenance” where the current degradation level is assumed to be lowered through an age reduction maintenance model, then these two alternative models provide different condition-based distributions of the degradation growth. Hence, in this latter case, a model misidentification can produce wrong predictions of the degradation growth and wrong estimates of the residual reliability. The discussed issue is illustrated via a numerical application based on a real set of degradation data.