Likelihood Inference for a Functional Marked Point Process with Cox-Ingersoll-Ross Process Marks

Abstract: This paper considers maximum likelihood inference for a functional marked point process - the stochastic growth-interaction process - which is an extension of the spatio-temporal growth-interaction process to the stochastic mark setting. As a pilot study we here consider a particular version of this extended process, which has a homogenous Poisson process as unmarked point process and shifted independent Cox-Ingersoll-Ross processes as functional marks. These marks have supports determined by the lifetimes generated by an immigration-death process. By considering a (temporally) discrete sample scheme for the marks and by considering the process' alternative evolutionary representation as a multivariate diffusion (Markovian) with jumps, the likelihood function is expressed as a product of the process' closed form transition densities. Additionally, under the assumption that the mark processes are started in their common stationary distribution, and under some restrictions on the underlying parameters, consistency and asymptotic normality of the maximum likelihood (ML) estimators are proved. The ML-estimators derived from the stationarity assumption are then compared numerically to the ML-estimators derived under non-stationarity, in order to investigate the robustness of the stationarity assumption. To illustrate the model's use in forestry, it is fitted to a data set of Scots pines.