Generalized method of moments for an extended gamma process

In reliability theory, a widely used process to model the phenomena of the cumulative deterioration of a system over time is the standard Gamma process. Based on several restrictions, such as a constant variance-to-mean ratio, this process is not always a suitable choice to describe the deterioration. A way to overcome these restrictions is to use an extended version of the Gamma process introduced by Cinlar (1980). In this paper, the aim is to propose statistical methods which enable us to estimate the unknown parameters of an EGP from a parametric form of the shape and scale functions. We here develop a generalized method of moments, which was introduced by Hansen (1982), based on the moments or the Laplace transform to estimate the parameters. Asymptotic properties are provided and the performance of the proposed estimation methods is illustrated on simulated data.