Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T14:14:45.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally asymptotic normality of Gibbs models on a lattice

Published online by Cambridge University Press:  01 July 2016

Shigeru Mase
Shigeru Mase*
Affiliation:
Hiroshima University
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the statistical estimation problem of potential functions of Gibbs models on the plane lattice. We assume that the area on which a random point pattern is observed is sufficiently large and take an asymptotic point of view. The main result is to show the locally asymptotic normality of the Gibbs model under certain conditions. From this result we can show the optimality of the maximum likelihood estimator employing known results about locally asymptotic normal families, though a practical computation of the maximum likelihood estimator presents difficulties. An estimation procedure based on the method of moments is also proposed.

Keywords

PLANE LATTICEASYMPTOTIC THEORYPOTENTIAL FUNCTIONMAXIMUM LIKELIHOOD ESTIMATORMOMENT ESTIMATOR
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 3 , September 1984 , pp. 585 - 602
Copyright
Copyright © Applied Probability Trust 1984 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was supported in part by a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture under Contract Number 321-6061-56530009.

References

Albeverio, S., Hoegh-Krohn, R. and Olsen, G. (1981) The global Markov property for lattice systems. J. Multivariate Anal. 11, 599–607.CrossRefGoogle Scholar
Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. J.R. Statist. Soc. B 36, 192–235.Google Scholar
Davies, R. B. (1973) Asymptotic inference in stationary Gaussian time series. Adv. Appl. Prob. 5, 469–497.CrossRefGoogle Scholar
Demongeot, J. (1981) Asymptotic inference for Markov random fields on Z2. In Numerical Methods in the Study of Critical Phenomena, Proc. Symp. Carry-le-Rouet, France, 1980, ed. Della Dora, J., Demongeot, J. and Lacolle, B. Springer Series in Synergetics 9, Springer-Verlag, Berlin.Google Scholar
Dobrushin, R. L. (1968a) The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Prob. Appl. 13, 197–224.CrossRefGoogle Scholar
Dobrushin, R. L. (1968b) Gibbsian random fields for lattice systems with pairwise interactions. Functional Anal. Appl. 2, 31–43.Google Scholar
Dobrushin, R. L. (1968C) The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Functional Anal. Appl. 2, 45–57.Google Scholar
Dobrushin, R. L. (1969) Gibbsian random fields, the general case. Functional Anal. Appl. 3, 27–35.CrossRefGoogle Scholar
Glötzl, E. (1980) On the statistics of Gibbsian processes. In Proc. 6th Internat. Conf., Wisla 1978, ed. Klonecki, W. et al. Lecture Notes in Statistics 2, Springer-Verlag, Berlin, 83–93.Google Scholar
Le Cam, L. (1960) Locally asymptotically normal families of distributions. Univ. Calif. Publ. Statist. 3, 37–98.Google Scholar
Nguyen, X. X. and Zessin, H. (1979a) Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.CrossRefGoogle Scholar
Nguyen, X. X. and Zessin, H. (1979b) Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105–115.Google Scholar
Pickard, D. K. (1976) Asymptotic inference for an Ising lattice. J. Appl. Prob. 13, 486–497.CrossRefGoogle Scholar
Pickard, D. K. (1977) Asymptotic inference for an Ising lattice, II. Adv. Appl. Prob. 9, 476–501.CrossRefGoogle Scholar
Pickard, D. K. (1979) Asymptotic inference for an Ising lattice, III, Non-zero field and ferromagnetic states. J. Appl. Prob. 16, 12–24.CrossRefGoogle Scholar
Pickard, D. K. (1982) Inference for general Ising models. Adv. Appl. Prob. 14, 345–357.CrossRefGoogle Scholar
Preston, C. J. (1974) Gibbs States on Countable Sets. Cambridge University Press.CrossRefGoogle Scholar
Preston, C. J. (1976) Random Fields. Lecture Notes in Mathematics 534, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Ripley, B. D. (1981) Spatial Statistics. Wiley, New York.CrossRefGoogle Scholar
Roussas, G. G. (1972) Contiguity of Probability Measures. Cambridge University Press.CrossRefGoogle Scholar
Strasser, H. (1978) Global asymptotic properties of risk functions in estimation. Z. Wahrscheinlichkeitsth. 45, 35–48.CrossRefGoogle Scholar
Strauss, D. J. (1975) Analysing binary lattice data with the nearest-neighbour property. J. Appl. Prob. 12, 702–712.CrossRefGoogle Scholar
Sylvester, G. S. (1979) Weakly coupled Gibbs measures. Z. Wahrscheinlichkeitsth. 50, 97–118.CrossRefGoogle Scholar
Zacks, S. (1971) The Theory of Statistical Inference. Wiley, New York.Google Scholar