Maximum Likelihood Estimates for Spatial Pure Birth Processes

We can write the likelihood function for the non-homogeneous pure birth process on a compact set through the method of projection introduced by Kurtz (1989) and studied by Garcia (1995), as the projection of the likelihood function. The fact that the projected likelihood can be interpreted as an expectation suggests that MCMC stochastic approximation could be useful in computing parameter estimates. In this case, we obtain a.s. convergence and distributional results for the convergence of the approximants to the maximum likelihood estimator.