Janžura, Martin Estimating interactions in binary lattice data with nearest-neighbor property. (English) Zbl 0668.62059 Kybernetika 23, 136-142 (1987). The estimation problem for the interaction parameter \(U\in R^ 3\) of a two-state Markov stationary Gibbs random field on \(({\mathbb{Z}}^+)^ 2\) is considered. Let \(x_{D(m,n)}\) be an observation of the field on a rectangular domain D(m,n) of size \(m\times n\), and put \(\hat U_{m,n}\) for the empirical minimum estimator for U from theorem 3.12 of the well- known book of D. Ruelle [Thermodynamic formalism. The mathematical structures of classical equilibrium. Statistical mechanics. (1978; Zbl 0401.28016)]. Theorem: \(\hat U_{m,n}\to U\) almost sure for m,n\(\to \infty\). An approximate calculation method for the values of \(\hat U_{m,n}\) through \(x_{D(m,n)}\), and an example are given. Reviewer: E.I.Trofimov Cited in 3 Documents MSC: 62M05 Markov processes: estimation; hidden Markov models 62M99 Inference from stochastic processes 82B30 Statistical thermodynamics 80A10 Classical and relativistic thermodynamics Keywords:two-state Markov stationary Gibbs random field; empirical minimum estimator PDF BibTeX XML Cite \textit{M. Janžura}, Kybernetika 23, 136--142 (1987; Zbl 0668.62059) Full Text: EuDML References: [1] J. E. Besag: On the statistical analysis of nearest-neighbor systems. Proceedings 9. European meeting of statisticians, Budapest 1972. · Zbl 0311.60028 [2] N. Dunford, J. T. Schwartz: Linear Operators, I. Interscience, New York 1958. · Zbl 0084.10402 [3] M. Janžura: Estimating interactions in binary data sequences. Kybernetika 22 (1986), 5, 377-384. · Zbl 0629.60104 · eudml:27732 [4] D. H. Mayer: The Ruelle-Araki Transfer Operator in Classical Mechanics. (Lecture Notes in Physics 123.) Springer-Verlag, Berlin-Heidelberg-New York 1980. · Zbl 0454.60079 [5] D. Ruelle: Thermodynamic Formalism. Addison Wesley, Reading, Mass. 1978. · Zbl 0401.28016 [6] D. Simon: A remark on Dobrushin’s uniqueness theorem. Comm. Math. Phys. 68 (1979), 183-185. · Zbl 0435.60099 · doi:10.1007/BF01418127 [7] D. J. Strauss: Analysing binary lattice data with the nearest-neighbor property. J. Appl. Probab. 72 (1975), 702-712. · Zbl 0322.62072 · doi:10.2307/3212721 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.