The transformed gamma process for degradation phenomena in presence of unexplained forms of unit-to-unit variability

The transformed gamma process is a suitable model for degradation phenomena where damages accumulate gradually over time in a sequence of tiny increments. Attractiveness of the transformed gamma process mainly lies in the fact that it provides a relatively easy way for dealing with phenomena in which the degradation increments over disjoint time intervals are not independent. The transformed gamma process is also a very flexible model. In fact, it is indexed by 2 functions, the "age function" and the "state function," whose mathematical form can be chosen ad hoc for modeling the dependence of the future degradation increment of a unit on its current age and state, respectively. In this paper, the transformed gamma process is adopted to describe the degradation paths of degrading units in the presence of an unexplained form of unit-to-unit variability. The degradation path of each unit is described via a transformed gamma process. Heterogeneity among paths of different units is accounted for by assuming that the scale parameters of the age and state functions vary randomly from unit to unit. Under these assumptions, a quite mathematically tractable model is obtained. The main properties of the proposed model are discussed, and inferential procedures based on the maximum likelihood criterion are implemented. A simple test is presented to check the goodness of fit of the proposed model. Three applicative examples, based on real degradation data, are developed.