Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model

TL;DR

This paper investigates the asymptotic behavior of the maximum pseudo-likelihood estimator for stationary Gibbs point processes, including Lennard-Jones models, establishing conditions for consistency and normality based on local energy functions.

Contribution

It provides general conditions for strong consistency and asymptotic normality of the estimator without requiring local stability or linearity in parameters.

Findings

Consistency conditions are satisfied for Lennard-Jones models.

Asymptotic normality conditions hold only for finite range Lennard-Jones models.

Results apply to a broad class of Gibbs point processes.

Abstract

This paper presents asymptotic properties of the maximum pseudo-likelihood estimator of a vector $\Vect{\theta}$ 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.