RESEARCH ARTICLE The transformed gamma process for degradation phenomena in presence of unexplained forms of unit‐to‐unit variability Massimiliano Giorgio 1 | Maurizio Guida 2,3 | Gianpaolo Pulcini 3 1 Department of Industrial and Information Engineering, Università della Campania “Luigi Vanvitelli”, Aversa, CE, Italy 2 Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Fisciano, SA, Italy 3 Istituto Motori, National Research Council (CNR), Naples, Italy Correspondence M. Giorgio, Department of Industrial and Information Engineering, Università della Campania “Luigi Vanvitelli,” Via Roma 29, 81031 Aversa (CE), Italy. Email: massimiliano. giorgio@unicampania.it Funding information Ministero dell'Istruzione, dell'Università e della Ricerca (Italy), Grant/Award Num- ber: PON03PE_00159_6 Abstract The transformed gamma process is a suitable model for degradation phenom- ena where damages accumulate gradually over time in a sequence of tiny incre- ments. 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 mathemati- cal form can be chosen ad hoc for modeling the dependence of the future deg- radation 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 ran- domly from unit to unit. Under these assumptions, a quite mathematically trac- table model is obtained. The main properties of the proposed model are discussed, and inferential procedures based on the maximum likelihood crite- rion 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. KEYWORDS degradation processes, dependent increments, maximum likelihood estimation, random effects, transformed gamma process 1 | INTRODUCTION The transformed gamma (TG) process was initially pro- posed in Giorgio et al. 1 Such a process can be viewed as a generalization of the very popular gamma process, which is nested within it. In fact, the TG process can be obtained from the gamma process via a non‐linear state transformation. As the gamma process (see for example Abdel‐Hameed, 2 van Noortwijk 3 ), the TG process can be used to describe degradation phenomena caused by con- tinuous use, where degradation growth takes place gradu- ally over time in a sequence of tiny increments, such as wear, chemical corrosion, consumption, fatigue, and so on. Attractiveness of the TG process, with respect to widely adopted independent increments degradation pro- cesses, such as the gamma, the Wiener, and the inverse Received: 1 February 2017 Revised: 24 October 2017 Accepted: 26 December 2017 DOI: 10.1002/qre.2271 Qual Reliab Engng Int. 2018;1–20. Copyright © 2018 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/qre 1