The Annals of Statistics

Consistency of likelihood estimation for Gibbs point processes

David Dereudre and Frédéric Lavancier

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • [1] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47 601–619.
  • [2] Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd ed. Wiley, Chichester.
  • [3] Dereudre, D. (2009). The existence of Quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains. Adv. in Appl. Probab. 41 664–681.
  • [4] Dereudre, D. (2016). Variational principle for Gibbs point processes with finite range interaction. Electron. Commun. Probab. 21 Paper No. 10, 11.
  • [5] Dereudre, D. and Lavancier, F. (2016). Supplement to “Consistency of likelihood estimation for Gibbs point processes.” DOI:10.1214/16-AOS1466SUPP.
  • [6] Diggle, P. J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. (1994). On parameter estimation for pairwise interaction point processes. In International Statistical Review/Revue Internationale de Statistique 99–117.
  • [7] Diggle, P. J. and Gratton, R. J. (1984). Monte Carlo methods of inference for implicit statistical models. J. Roy. Statist. Soc. Ser. B 46 193–227.
  • [8] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
  • [9] Georgii, H.-O. (1994). Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Related Fields 99 171–195.
  • [10] Georgii, H.-O. (1995). The equivalence of ensembles for classical systems of particles. J. Stat. Phys. 80 1341–1378.
  • [11] Geyer, C. J. and Møller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21 359–373.
  • [12] Guyon, X. (1995). Random Fields on a Network: Modeling, Statistics, and Applications. Springer, New York.
  • [13] Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Wiley, Chichester.
  • [14] Jensen, J. L. (1993). Asymptotic normality of estimates in spatial point processes. Scand. J. Statist. 20 97–109.
  • [15] Kendall, W. S., van Lieshout, M. N. M. and Baddeley, A. J. (1999). Quermass-interaction processes: Conditions for stability. Adv. in Appl. Probab. 31 315–342.
  • [16] Künsch, H. (1981). Thermodynamics and statistical analysis of Gaussian random fields. Z. Wahrsch. Verw. Gebiete 58 407–421.
  • [17] Mase, S. (1992). Uniform LAN condition of planar Gibbsian point processes and optimality of maximum likelihood estimators of soft-core potential functions. Probab. Theory Related Fields 92 51–67.
  • [18] Mase, S. (2002). Asymptotic properties of MLE’s of Gibbs models on $\mathbb{R}^{d}$. Unpublished manuscript.
  • [19] Mateu, J. and Montes, F. (2001). Likelihood inference for Gibbs processes in the analysis of spatial point patterns. Int. Stat. Rev. 69 81–104.
  • [20] Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall, Boca Raton.
  • [21] Ogata, Y. and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. J. Roy. Statist. Soc. Ser. B 46 496–518.
  • [22] Pickard, D. K. (1987). Inference for discrete Markov fields: The simplest nontrivial case. J. Amer. Statist. Assoc. 82 90–96.
  • [23] Preston, C. (1976). Random Fields. Springer, Berlin.
  • [24] Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. Benjamin, New York.
  • [25] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127–159.
  • [26] Van Lieshout, M. N. M. (2000). Markov Point Processes and Their Applications. World Scientific, Singapore.
  • [27] Varadhan, S. R. S. (1988). Large deviations and applications. In École D’Été de Probabilités de Saint-Flour XV–XVII, 198587. Lecture Notes in Math. 1362 1–49. Springer, Berlin.
  • [28] Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52 1670–1684.

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.