On estimation of the mean and covariance parameter for Gaussian random fields

In the paper we consider the problem of estimation for a multivariate Gaussian random field whose vector mean depends linearly of a multidimensional parameter theta. A Bayesian estimator is derived and its distributions are analyzed under the assumption that the covariance Function is completely known. Further, an invariant structure is introduced and the optimal invariant terminal decision function obtained. Finally, assuming that the covariance function is known up to a parameter sigma(2) we estimate the parameters theta and sigma(2) and prove some of their properties. The results of Ibarrola and Velez (1990), (1992), for multidimensional stochastic processes are extended to the the present context and similar results are obtained.