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Asymptotic results in parameter estimation for Gibbs random fields. (English) Zbl 0962.62092
The author studies asymptotic properties of maximum likelhood (ML) estimators and a class of maximum pseudo-likelihood (MPL) estimators for parameters of stationary Gibbs random fields. The strong consistency of the ML and particular MPL estimators is proved. It is shown that the ML estimator can be approximated by a sequence of MPL estimators for every fixed sample size. The asymptotic normality of MPL estimators is proved provided the true Gibbs distribution is ergodic; for a stationary Gibbs distribution the limiting distribution of MPL estimators is mixed normal. The asymptotic normality and efficiency of ML estimators is proved inside the Dobrushin uniqueness region. In this case MPL estimators are less efficient compared to ML estimators. However, it is shown that the maximum asymptotic efficiency is approached by MPL estimators with growing ranges.

62M40 Random fields; image analysis
62F12 Asymptotic properties of parametric estimators
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