Parameter estimation for a stationary process on a d-dimensional lattice

Abstract

We study asymptotic properties of various estimation procedures for a general stationary regular process on a d-dimensional lattice. Differences between d = 1, time series, and d ≥ 2 spatial processes, are pointed out. We suppose that the process is observed on a set PN, with cN points, which tends to infinity with the same speed in all directions. The relative edge effect is of order cN−1/d, increasing with d: this effect is without statistical importance if d = 1, but is important if d ≥ 2. We give a √cN-consistent, asymptotically normal estimator of the underlying parameter, the procedure being constructed by a modification of Whittle's approximation to the log likelihood. In the Gaussian case, this procedure is asymptotically efficient.