The discretely observed immigration-death process: Likelihood inference and spatiotemporal applications

ABSTRACT

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.

Keywords:

• Asymptotic normality
• Consistency
• Homogenous spatial immigration-death process
• Maximum likelihood
• Spatial birth–death process
• Spatiotemporal growth-interaction process

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H11
• 62M30
• 62H12
• 62M05
• 62F10
• 62F12

Acknowledgments

The authors would like to thank Aila Särkkä (Chalmers University of Technology), Bo Ranneby, and the late Lennart Norell (Swedish University of Agricultural Sciences) for useful comments. The authors would also like to thank the two anonymous referees for their comments and feedback.

Funding

This research has been supported by the Swedish Research Council and the Swedish Foundation for Strategic Research.