A transformed gamma process for bounded degradation phenomena

Most of the stochastic models adopted to describe the evolution over time of degradation phenomena of technological units assume that their degradation level can increase indeterminately. However, these degradation phenomena are typically subjected to obvious bounds, if only because technological units have finite size. In fact, very often, this inconsistency does not significantly affect the effectiveness of unbounded degradation models, since degrading units are usually assumed to fail when their degradation level exceeds a failure threshold that is much smaller than the obvious bounds. Nevertheless, in some cases, due to the very nature of the underlying degradation mechanism, less obvious bounds could exist, which are not necessarily far from the failure thresholds. The question that arises is whether the use of a bounded degradation model, in this latter type of experimental situations, could be beneficial. For this purpose, since a bounded degradation process should necessarily have dependent increments, in this paper we investigate the potential of a new bounded transformed gamma (TG) process to adequately describe bounded degradation phenomena and predict their future evolution. Differently from other existing gamma process based bounded degradation models, here the upper bound is treated as an unknown parameter that has to be estimated from the available degradation data. A numerical example is presented where the parameters of the proposed model are estimated from simulated data. Then the model is applied to a set of wear measures of cylinder liners that equip a diesel engine for marine propulsion, which have also stimulated this study. Model parameters are estimated by using the maximum likelihood (ML) method. The fitting ability of the proposed new bounded process is compared to that of an unbounded gamma process, which was previously adopted to analyze the same liner wear data. Obtained results are critically discussed in the paper.