A Bounded Transformation of the Gamma Degradation Process

Although the finiteness of physical dimensions and/or the nature of the degradation mechanism make the degradation phenomena of several technological units naturally asymptotically bounded, the stochastic models used to describe these 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 is slightly greater than the threshold value. The aim of this paper is then to propose an asymptotically bounded transformation of the gamma process able to correctly model the bounded degradation phenomena even when the "natural" bound and the threshold have comparable values. This idea is not completely new, but, unlike what is assumed in existing models, the upper bound is here treated as an unknown parameter that must be estimated from the available data. The proposed approach is then applied to a real dataset consisting of the wear measurements of eight cylinder liners equipping a Diesel engine for marine propulsion. Model parameters are estimated by using the maximum likelihood method. The fitting ability of the proposed bounded process is compared with that of the unbounded transformed gamma process, previously adopted to analyze these wear data. A condition-based maintenance policy is also applied to the above wear data in order to highlight the need to correctly model the degradation phenomena for avoiding unnecessary maintenance costs. Potentiality of the proposed approach are critically discussed in the paper.