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Estimating interactions in binary lattice data with nearest-neighbor property. (English) Zbl 0668.62059
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

62M05 Markov processes: estimation; hidden Markov models
62M99 Inference from stochastic processes
82B30 Statistical thermodynamics
80A10 Classical and relativistic thermodynamics
Full Text: EuDML
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