ON MAXIMUM LIKELIHOOD ESTIMATION FOR GAUSSIAN SPATIAL AUTOREGRESSION MODELS

JAROSLAV MOHAPL

Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(Received September 25, 1996; revised February 17, 1997)

Abstract. The article presents a central limit theorem for the maximum likelihood estimator of a vector-valued parameter in a linear spatial stochastic difference equation with Gaussian white noise right side. The result is compared to the known limit theorems derived for the approximate likelihood e.g. by Whittle (1954, Biometrika, 41, 434–439), Guyon (1982, Biometrika, 69, 95–105) and Rosenblatt (1985, Stationary Sequences and Random Fields, Birkhäuser, Boston) and to the asymptotic properties of the quasi-likelihood studied by Heyde and Gay (1989, Stochastic Process. Appl., 31, 223–236; 1993, Stochastic Process. Appl., 45, 169–182). Application of the theory is demonstrated on several classes of models including the one considered by Niu (1995, J. Multivariate Anal., 55, 82–104).

Key words and phrases: Spatial process, asymptotic normality, consistency, lattice sampling, stochastic difference equation.