Modified Gaussian likelihood estimators for ARMA models on ZdZd

For observations from an auto-regressive moving-average process on any number of dimensions, we propose a modification of the Gaussian likelihood, which when maximized corrects the edge-effects and fixes the order of the bias for the estimators derived. We show that the new estimators are not only consistent but also asymptotically normal for any dimensionality. A classical one-dimensional, time series result for the variance matrix is established on any number of dimensions and guarantees the efficiency of the estimators, if the original process is Gaussian. We have followed a model-based approach and we have used finite numbers for the corrections per dimension, which are especially made for the case of the auto-regressive moving-average models of fixed order.