Abstract
This paper presents asymptotic properties of the maximum pseudo-likelihood estimator of a vector parameterizing a stationary Gibbs point process. Sufficient conditions, expressed in terms of the local energy function defining a Gibbs point process, to establish strong consistency and asymptotic normality results of this estimator depending on a single realization, are presented. These results are general enough to no longer require the local stability and the linearity in terms of the parameters of the local energy function. We consider characteristic examples of such models, the Lennard-Jones and the finite range Lennard-Jones models. We show that the different assumptions ensuring the consistency are satisfied for both models whereas the assumptions ensuring the asymptotic normality are fulfilled only for the finite range Lennard-Jones model.
Highlights
These last years, much attention has been paid to spatial point pattern data, and especially to models and methodologies for fitting them, see Møller (2008) for a recent overview of this topic and Daley and Vere-Jones (1988), Stoyan et al (1987) Møller and Waagepetersen (2003) or Illian et al (2008) for more general information
A Gibbs point process is defined through its probability measure having a Radon-Nykodym derivative with respect to a Poisson point process measure proportional to e−V (φ) where V (φ) corresponds to the energy function of the configuration of points φ
In Billiot et al (2008), we obtain consistency and asymptotic normality for exponential family models of Gibbs point processes, that is, on models with energy functions that are linear in terms of the parameters
Summary
These last years, much attention has been paid to spatial point pattern data, and especially to models and methodologies for fitting them, see Møller (2008) for a recent overview of this topic and Daley and Vere-Jones (1988), Stoyan et al (1987) Møller and Waagepetersen (2003) or Illian et al (2008) for more general information. Besag et al (1982) further considered this method for pairwise interaction point processes, and Jensen and Møller (1991) extended the definition of the pseudo-likelihood function to the general class of marked Gibbs point processes. In Billiot et al (2008), we obtain consistency and asymptotic normality for exponential family models of Gibbs point processes, that is, on models with energy functions that are linear in terms of the parameters. For general Gibbs point processes, sufficient conditions, expressed in terms of the local energy function to establish strong consistency and asymptotic normality results of this estimator are presented.
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