Degradation processes: a case where statistical tools are unable to choose between two models based on different physical assumptions

The gamma process is often used to describe degradation processes where the degradation growth does not depend on the current degradation level but at most on the current age. In particular, the gamma process with random scale parameter is often used to describe degradation phenomena in presence of random effects. In this paper, we introduce a new degradation process, the Transformed Beta (TB) process, where the degradation growth depends on both the current age and degradation level. Then, we show that the likelihood function relative to degradation data under the assumption that the degradation process is gamma with gamma-distributed (random) scale parameter, is the same as the likelihood function under the assumption that the process is TB without random effects. In particular, we show that the distribution of the degradation increment during a future time interval under the above two alternatives is the same, and hence the residual reliability and the mean residual lifetime are the same. However, if the unit is subject to a so-called "imperfect maintenance", where the current degradation level is assumed to be lowered through an age reduction model, then the distribution of the degradation growth under the two alternatives is no longer the same, and hence an inappropriate degradation model produces wrong estimates of residual reliability and mean lifetime. Finally, a numerical application shows the errors produced by a wrong model when the unit is imperfectly maintained.