• Bernoulli
  • Volume 15, Number 4 (2009), 1368-1396.

Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes

David Dereudre and Frédéric Lavancier

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In this paper, we study Gibbs point processes involving a hardcore interaction which is not necessarily hereditary. We first extend the famous Campbell equilibrium equation, initially proposed by Nguyen and Zessin [Math. Nachr. 88 (1979) 105–115], to the non-hereditary setting and consequently introduce the new concept of removable points. A modified version of the pseudo-likelihood estimator is then proposed, which involves these removable points. We consider the following two-step estimation procedure: first estimate the hardcore parameter, then estimate the smooth interaction parameter by pseudo-likelihood, where the hardcore parameter estimator is plugged in. We prove the consistency of this procedure in both the hereditary and non-hereditary settings.

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Bernoulli, Volume 15, Number 4 (2009), 1368-1396.

First available in Project Euclid: 8 January 2010

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Campbell measure consistency Gibbs point process non-hereditary interaction pseudo-likelihood estimator spatial statistics


Dereudre, David; Lavancier, Frédéric. Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes. Bernoulli 15 (2009), no. 4, 1368--1396. doi:10.3150/09-BEJ198.

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  • [1] Bertin, E., Billiot, J.M. and Drouilhet, R. (1999). Existence of nearest-neighbors spatial Gibbs models. Adv. in Appl. Probab. 31 895–909.
  • [2] Billiot, J.-M., Coeurjolly, J.-F. and Drouilhet, R. (2008). Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes. Electron. J. Statist. 2 234–254.
  • [3] Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 617–666.
  • [4] Berthelsen, K.K. and Moller, J. (2003). Likelihood and non-parametric Bayesian MCMC inference for spatial point processes based on perfect simulation and path sampling. Scand. J. Statist. 30 549–564.
  • [5] Dereudre, D. (2008). Gibbs Delaunay tessellations with geometric hardcore conditions. J. Statist. Phys. 131 127–151.
  • [6] Georgii, H.-O. (1979). Canonical Gibbs Measure. Lecture Notes in Math. 760. Berlin: Springer.
  • [7] Geyer, C.J. and Thompson, E.A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 657–699.
  • [8] Guyon, X. (1995). Random Fields on a Network. New York: Springer.
  • [9] Jensen, J.L. and Künsch, H.R. (1994). On asymptotic normality of pseudo-likelihood estimates for pairwise interaction process. Ann. Inst. Statist. Math. 46 475–486.
  • [10] Jensen, J.L. and Moller, J. (1991). Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1 445–461.
  • [11] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Process. Chichester: Wiley.
  • [12] Preston, C. (1976). Random Fields. Lecture Notes in Math. 534. Berlin: Springer.
  • [13] Propp, J.G. and Wilson, D.B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223–252.
  • [14] Mase, S. (1995). Consistency of maximum pseudo-likelihood estimator of continuous state space Gibbsian process. Ann. Appl. Probab. 5 603–612.
  • [15] Moller, J. and Waggepertesen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Boca Raton: Chapman and Hall.
  • [16] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127–159.
  • [17] Nguyen, X.X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88 105–115.