On consistency of a class of estimators for exponential families of Markov random fields on the lattice (Q1192982)

A statistical estimation problem is considered for parametric exponential non-Gaussian families of Markov random fields, and more generally, Gibbs distributions (GD). Consistency, exponential convergence rates and Bahadur optimality are proved for the maximum and pseudo-maximum likelihood estimators under conditions allowing nonergodic, nonstationary and phase transition cases. The proofs are based on the large deviations estimates for the GD obtained by \textit{the author} [C. R. Acad. Sci., Paris, Ser. I 303, 511- 513 (1986; Zbl 0606.60035)], \textit{H. Föllmer} and \textit{S. Orey} [Ann. Probab. 16, No. 3, 961-977 (1988; Zbl 0648.60028)] and by \textit{S. Olla} [Probab. Theory Relat. Fields 77, No. 3, 343-357 (1988; Zbl 0621.60031)].