Summary
A parametric model of planar point patterns in a bounded region is constructed using grand canonical Gibbsian point processes with soft-core potential functions. A simple and explicit condition that this model becomes a uniform locally asymptotic normal (ULAN) family will be given. From this result we can conclude that the maximum likelihood estimator of the potential function is asymptotically efficient for a wide class of loss functions.
Article PDF
Similar content being viewed by others
References
Besag, J., Milne, R., Zachary, S.: Point process limits of lattice processes. J. Appl. Probab.19, 210–216 (1982)
Diggle, P.J., Gates, D.J., Stibbard, A.: A nonparametric estimator for pairwise-interaction point processes. Biometrika74, 763–770 (1987)
Fiksel, T.: Estimation of interaction potentials of Gibbsian point processess. Operationsforsch. Stat. Ser. Stat.20, 270–278 (1984)
Geman, S., Geman, D.: Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Trans. PAMI6, 721–741 (1984)
Gidas, B.: Parameter estimation for Gibbs distributions, I: Fully observed data. In: Chellapa, R., Jain, A. (eds.) Markov random fields; theory and applications. New York: Academic Press 1991
Glötzl, E. and Rauschenschwandtner, B.: On the statistics of Gibbsian processes. In: Re've'sz, P. et al. (eds.) The first Pannonian symposium on mathematical statistics, Bad Tatzmannsdorf, Austria, 1979. (Lect. Notes Stat., vol. 8, pp. 83–93) Berlin Heidelberg New York: Springer 1981
Grandy, Jr., W.T.: Foundations of Statistical Mechanics. Volume I: Equilibrium theory. Dordrecht: Reidel 1987
Hanisch, K.-H., Stoyan, D.: Remarks on statistical inference and prediction for a hard-core clustering model. Math. Operationsforsch. Stat. Ser. Stat.14, 559–567 (1983)
Hörmander, L.: An introduction to complex analysis in several variables. Amsterdam: North-Holland 1973
Ibragimov, I.A., Has'minskii, R.Z.: Statistical estimation, asymptotic theory. Berlin Heidelberg New York: Springer 1981
Jensen, J.L.: Asymptotic normality of estimates in spatial point processes. Research Reports No. 210, Department of Theoretical Statistics, University of Aarhus (1990)
Ji, C.: Estimating Functionals of one-dimensional Gibbs states. Probab. Theory Relat. Fields82, 155–175 (1989)
Kutoyants, Yu.A.: Parameter estimation for stochastic processes. Berlin: Heddermann 1984
Mase, S.: Locally asymptotic normality of Gibbs models on a lattice. J. Appl. Probab.16, 585–602 (1984)
Mase, S.: Asymptotic equivalence of grand canonical MLE and canonical MLE of pair potential functions of Gibbsian point process model. (Submitted, 1991)
Minlos, R.A., Pogosian, S.K.: Estimates of ursell functions, group functions, and their derivatives. Theor. Math. Phys.31, 408–418 (1977)
Moyeed, R.A., Baddeley, A.J.: Stochastic approximation of the MLE for a spatial point pattern. Report BS-R8926, Center for Mathematics and Computer Sciences, Stichting Mathematisch Centrum. (1989)
Ogata, Y., Tanemura, M.: Approximation of likelihood functions in estimating interaction potentials from spatial point patterns. J.R. Stat. Soc. Ser.B46, 496–518 (1984)
Ogata, Y., Tanemura, M.: Estimation of interaction potentials of marked spatial point processes through the maximum-likelihood method. Biometrics41, 421–433 (1985)
Ogata, Y., Tanemura, M.: Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann. Inst. Stat. Math.B33, 315–338 (1981)
Penttinen, A.: Modelling interactions in spatial point patterns: parameter estimation by the maximum-likelihood method. Jyväskyla Stud. Comp. Sci., Econ. Stat.7, 1–107 (1984)
Pickard, D.K.: Asymptotic inference for an Ising lattice. J. Appl. Probab.13, 486–497 (1976)
Pickard, D.K.: Asymptotic inference for an Ising lattice, II. Adv. Appl. Probab.9, 476–501 (1977)
Pickard, D.K.: Asymptotic inference for an Ising lattice, III, Non-zero field and ferromagnetic states. J. Appl. Probab.16, 12–24 (1979)
Pickard, D.K.: Inference for general Ising models. Adv. Appl. Probab.14, 345–357 (1982)
Pogosian, S.K.: Asymptotic expansion in the local limit theorem for the particle number in the grand canonical ensemble. In: Fritz, J. et al. (eds.) Random Fields. vol. 2, pp. 889–913. Colloquia Mathematica Societatis János Bolyai. Amsterdam: North-Holland 1981
Pogosian, S.K.: Asymptotic expansion of the logarithm of the partition function. Commun. Math. Phys.95, 227–245 (1984)
Preston, C.: Random fields. Berlin Heidelberg New York: Springer 1976
Ripley, B.D.: Modelling spatial patterns (with discussion). J. R. Stat. Soc. Ser.B 39, 172–212 (1977)
Ripley, B.D.: Statistical inference for spatial processes. Cambridge: Cambridge University Press 1988
Rodgers, G.S.: Matrix derivatives. New York: Marcel Dekker 1980
Ruelle, D.: Statistical mechanics rigorous results. New York: Benjamin 1969
Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic geometry and its applications. New York: Wiley 1987
Takacs, R.: Estimator for the pair-potential of a Gibbsian point process. Statistics17, 429–433 (1986)
Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transition, 1. theory of condensation. Phys. Rev.87, 404–409 (1952)
Younes, L.: Parametric inference for imperfectly observed Gibbsian fields. Probab. Theory Relat. Fields82, 625–645 (1989)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mase, S. Uniform LAN condition of planar Gibbsian point processes and optimality of maximum likelihood estimators of soft-core potential functions. Probab. Th. Rel. Fields 92, 51–67 (1992). https://doi.org/10.1007/BF01205236
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01205236