About Bounded Transformations of the Gamma Degradation Process

Although the degradation processes of technological units are naturally bounded, due to the finiteness of their physical dimensions and/or the nature itself of the degradation mechanism, the models adopted to describe degradation phenomena are typically unbounded. In general, this apparent contradiction does not significantly affect the effectiveness of unbounded degradation models, because degrading units are conventionally considered failed when their degradation level exceeds a threshold value that is quite far from the "natural" bounds. On the other side, however, the effectiveness of an unbounded degradation models can drastically diminish if the physical bound and threshold have comparable values. The aim of this paper is then to investigate the potentiality of the transformed gamma process in modelling bounded degradation phenomena. This idea is not new. Yet, differently than in other existing models, here the upper bound is treated as an unknown parameter and is estimated from the available degradation data. The proposed approach, which led to the definition of a bounded (state-dependent) transformed gamma process, is illustrated starting with a motivating example, which is developed on the basis of a real set of wear data of cylinder liners equipping a diesel engine for marine propulsion. Model parameters are estimated by using the maximum likelihood method. Fitting ability of the innovative proposed bounded process is compared with those of the unbounded gamma process, previously adopted to analyze these wear data. Potentiality of the proposed approach are critically discussed in the paper.