Time-invariance modeling and estimation for spatial point processes: General theory
نویسندگان
چکیده
Models for spatial point processes are characterized as the stationary distributions of spatial birth and death processes. The Markov chain Monte Carlo algorithm determined by the underlying birth and death process immediately gives a method of simulation, and the time-invariance method of estimation proposed by Baddeley (2000) gives a general method for deriving parameter estimates for the models. Typically, in applications of Markov chain Monte Carlo and time-invariance estimation, one begins with the model of interest specified in some other way and then constructs a Markov process having the desired distribution as its stationary distribution; however, specifying the model directly in terms of the Markov process provides an intuitive and flexible approach to modeling. In time-invariance estimation, the parameter estimates are obtained by equating to zero the generator of the Markov process applied to a suitable collection of statistics. The estimators depend on the choice of the statistics, and the art is to find statistics that give estimators that are easy to compute and have good statistical properties. Statistics are given that are useful for spatial point processes including Gibbs and nonGibbs models, and a large sample limit theorem is proved for these statistics which enables one to verify consistency of the estimators for particular classes of models. MSC 2000 subject classifications: Primary: 62M30, 60G55 Secondary: 62F12, 62F10, 60F15
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