The Annals of Statistics

Consistency of likelihood estimation for Gibbs point processes

David Dereudre and Frédéric Lavancier

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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 nonlinearly 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.

Article information

Source
Ann. Statist., Volume 45, Number 2 (2017), 744-770.

Dates
Received: April 2015
Revised: March 2016
First available in Project Euclid: 16 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1494921956

Digital Object Identifier
doi:10.1214/16-AOS1466

Mathematical Reviews number (MathSciNet)
MR3650399

Zentralblatt MATH identifier
1371.62021

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M30: Spatial processes 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Parametric estimation variational principle Strauss model Lennard–Jones model area-interaction model

Citation

Dereudre, David; Lavancier, Frédéric. Consistency of likelihood estimation for Gibbs point processes. Ann. Statist. 45 (2017), no. 2, 744--770. doi:10.1214/16-AOS1466. https://projecteuclid.org/euclid.aos/1494921956


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Supplemental materials

  • Supplement to “Consistency of likelihood estimation for Gibbs point processes”. This supplementary material provides the proofs of Lemma 4, Corollary 1, Theorems 2–4 and Propositions 1 and 2.