Consistency of likelihood estimation for Gibbs point processes

TL;DR

This paper proves the strong consistency of maximum likelihood estimators for a broad class of Gibbs point process models, including various interaction types and parameters, ensuring reliable statistical inference.

Contribution

It establishes the first general proof of MLE consistency for diverse Gibbs models, covering complex interactions and parameter dependencies.

Findings

MLE is strongly consistent for the Strauss model

MLE is strongly consistent for the Lennard-Jones model

MLE is strongly consistent for the area-interaction model

Abstract

Strong consistency of the maximum likelihood estimator (MLE) for parametric Gibbs point process models is established. The setting is very general. It includes pairwise pair potentials, finite and infinite multibody interactions and geometrical interactions, where the range can be finite or infinite. The Gibbs interaction may depend linearly or non-linearly on the parameters, a particular case being hardcore parameters and interaction range parameters. As important examples, we deduce the consistency of the MLE for all parameters of the Strauss model, the hardcore Strauss model, the Lennard-Jones model and the area-interaction model.