A Gamma Process with Three Sources of Variability

Degradation modeling requires to consider the complexity of both the internal structure of highly reliable products and the environmental conditions, to define appropriate models to obtain estimations about the reliability and quality. These conditions reflect different sources of variability that need to be considered in the aims of obtaining accurate estimations. Although several models have been proposed in the literature, few of them consider several simultaneous sources of variability. In this paper, we propose a model based on the gamma process that considers three sources of variability, specifically in the threshold, the initial level of degradation, and in the scale parameter of the gamma process. The model considers a convolution operation of the threshold and the initial level to then be characterized via numerical integration with the gamma process with random scale. The obtained results showed that the model can be used to model the degradation of products with these sources of variability, which means that it can used for case studies where both the initial level and threshold are inherently random and the randomness in the scale parameter can be proved. The performance is illustrated with a comprehensive simulation study and with the application in a case study