Bernoulli

  • Bernoulli
  • Volume 23, Number 2 (2017), 1299-1334.

Parametric estimation of pairwise Gibbs point processes with infinite range interaction

Jean-François Coeurjolly and Frédéric Lavancier

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Abstract

This paper is concerned with statistical inference for infinite range interaction Gibbs point processes, and in particular for the large class of Ruelle superstable and lower regular pairwise interaction models. We extend classical statistical methodologies such as the pseudo-likelihood and the logistic regression methods, originally defined and studied for finite range models. Then we prove that the associated estimators are strongly consistent and satisfy a central limit theorem, provided the pairwise interaction function tends sufficiently fast to zero. To this end, we introduce a new central limit theorem for almost conditionally centered triangular arrays of random fields.

Article information

Source
Bernoulli, Volume 23, Number 2 (2017), 1299-1334.

Dates
Received: July 2015
Revised: October 2015
First available in Project Euclid: 4 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1486177400

Digital Object Identifier
doi:10.3150/15-BEJ779

Mathematical Reviews number (MathSciNet)
MR3606767

Zentralblatt MATH identifier
06701627

Keywords
central limit theorem Lennard–Jones potential pseudo-likelihood

Citation

Coeurjolly, Jean-François; Lavancier, Frédéric. Parametric estimation of pairwise Gibbs point processes with infinite range interaction. Bernoulli 23 (2017), no. 2, 1299--1334. doi:10.3150/15-BEJ779. https://projecteuclid.org/euclid.bj/1486177400


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References

  • [1] Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014). Logistic regression for spatial Gibbs point processes. Biometrika 101 377–392.
  • [2] Baddeley, A. and Dereudre, D. (2013). Variational estimators for the parameters of Gibbs point process models. Bernoulli 19 905–930.
  • [3] Baddeley, A., Rubak, E. and Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. London: Chapman & Hall/CRC Press.
  • [4] Baddeley, A. and Turner, R. (2000). Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42 283–322.
  • [5] Baddeley, A. and Turner, R. (2005). Spatstat: An $\mathsf{R}$ package for analyzing spatial point patterns. J. Stat. Softw. 12 1–42.
  • [6] Bertin, E., Billiot, J.-M. and Drouilhet, R. (1999). Existence of “nearest-neighbour” spatial Gibbs models. Adv. in Appl. Probab. 31 895–909.
  • [7] Besag, J. (1977). Some methods of statistical analysis for spatial data. In Proceedings of the 41st Session of the International Statistical Institute (New Delhi, 1977), Vol. 2 47 77–91, 138–147. Bull. Inst. Internat. Statist., 2, With discussion.
  • [8] Billiot, J.-M., Coeurjolly, J.-F. and Drouilhet, R. (2008). Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes. Electron. J. Stat. 2 234–264.
  • [9] Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047–1050.
  • [10] Chiu, S.N., Stoyan, D., Kendall, W.S. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd ed. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [11] Coeurjolly, J.-F. and Drouilhet, R. (2010). Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard–Jones model. Electron. J. Stat. 4 677–706.
  • [12] Coeurjolly, J.-F. and Lavancier, F. (2013). Residuals and goodness-of-fit tests for stationary marked Gibbs point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 247–276.
  • [13] Coeurjolly, J.-F. and Møller, J. (2014). Variational approach for spatial point process intensity estimation. Bernoulli 20 1097–1125.
  • [14] Coeurjolly, J.-F. and Rubak, E. (2013). Fast covariance estimation for innovations computed from a spatial Gibbs point process. Scand. J. Stat. 40 669–684.
  • [15] Comets, F. and Janžura, M. (1998). A central limit theorem for conditionally centred random fields with an application to Markov fields. J. Appl. Probab. 35 608–621.
  • [16] Daley, D.J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd ed. Probability and Its Applications (New York). New York: Springer.
  • [17] Decreusefond, L. and Flint, I. (2014). Moment formulae for general point processes. J. Funct. Anal. 267 452–476.
  • [18] Dedecker, J. (1998). A central limit theorem for stationary random fields. Probab. Theory Related Fields 110 397–426.
  • [19] Dereudre, D. and Lavancier, F. (2009). Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes. Bernoulli 15 1368–1396.
  • [20] Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Comm. Math. Phys. 48 31–51.
  • [21] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics 9. Berlin: de Gruyter.
  • [22] Guyon, X. (1995). Random Fields on a Network: Modeling, Statistics, and Applications. Probability and Its Applications (New York). New York: Springer.
  • [23] Guyon, X. and Künsch, H.R. (1992). Asymptotic comparison of estimators in the Ising model. In Stochastic Models, Statistical Methods, and Algorithms in Image Analysis (Rome, 1990). Lecture Notes in Statist. 74 177–198. Berlin: Springer.
  • [24] Heinrich, L. (1992). Mixing properties of Gibbsian point processes and asymptotic normality of Takacs–Fiksel estimates. Preprint.
  • [25] Huang, F. and Ogata, Y. (1999). Improvements of the maximum pseudo-likelihood estimators in various spatial statistical models. J. Comput. Graph. Statist. 8 510–530.
  • [26] Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Statistics in Practice. Chichester: Wiley.
  • [27] Jensen, J.L. (1993). Asymptotic normality of estimates in spatial point processes. Scand. J. Stat. 20 97–109.
  • [28] Jensen, J.L. and Künsch, H.R. (1994). On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes. Ann. Inst. Statist. Math. 46 475–486.
  • [29] Jensen, J.L. and Møller, J. (1991). Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1 445–461.
  • [30] Lennard-Jones, J.E. (1924) On the determination of molecular fields. In Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 463–477. The Royal Society, London.
  • [31] Mase, S. (1995). Consistency of the maximum pseudo-likelihood estimator of continuous state space Gibbsian processes. Ann. Appl. Probab. 5 603–612.
  • [32] Mecke, J. (1968). Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrsch. Verw. Gebiete 11 74–81.
  • [33] Møller, J. and Waagepetersen, R.P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Boca Raton, FL: Chapman & Hall/CRC.
  • [34] Nguyen, X.-X. and Zessin, H. (1979). Ergodic theorems for spatial processes. Z. Wahrsch. Verw. Gebiete 48 133–158.
  • [35] Nguyen, X.-X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88 105–115.
  • [36] Ogata, Y. and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann. Inst. Statist. Math. 33 315–338.
  • [37] Preston, C. (1976). Random Fields. Lecture Notes in Mathematics 534. Berlin: Springer.
  • [38] Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. New York: W.A. Benjamin.
  • [39] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127–159.
  • [40] Slivnyak, I.M. (1962). Some properties of stationary flows of homogeneous random events. Theory Probab. Appl. 7 336–341.
  • [41] van Lieshout, M.N.M. (2000). Markov Point Processes and Their Applications. London: Imperial College Press.

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