Maximum Likelihood Estimation for an Inhomogeneous Gamma Process with a Log-linear Rate Function

Abstract

An inhomogeneous gamma process is a compromise between a renewal process and a nonhomogeneous Poisson process, since its failure probability at a given time depends both on the age of the system and on the distance from the last failure time. The inhomogeneous gamma process with a log-linear rate function is often used in modelling of recurrent event data. In this paper, it is proved that the suitably non-uniform scaled maximum likelihood estimator of the three-dimensional parameter of this model is asymptotically normal, but it enjoys the curious property that the covariance matrix of the asymptotic distribution is singular. A simulation study is presented to illustrate the behaviour of the maximum likelihood estimators in finite samples. Obtained results are also applied to real data analysis.