A Non-homogeneous Count Process: Marginalizing a Poisson Driven Cox Process

The paper considers a Cox process where the stochastic intensity function for the Poisson data model is itself a non-homogeneous Poisson process. We show that it is possible to obtain the marginal data process, namely a non-homogeneous count process exhibiting over-dispersion. While the intensity function is non-decreasing, it is straightforward to transform the data so that a non-decreasing intensity function is appropriate. We focus on a time series for arrival times of a process and, in particular, we are able to find an exact form for the marginal probability for the observed data, so allowing for an easy to implement estimation algorithm via direct calculations of the likelihood function.