We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data.

Communications in Statistics - Theory and Methods

Cronie, O., & Yu, J. (2016). The discretely observed immigration-death process: likelihood inference and spatiotemporal applications. Communications in Statistics - Theory and Methods, 45(18), 5279–5298. doi:10.1080/03610926.2014.942433