Correlated optimum design with parametrized covariance function. Justification of the Fisher information matrix and of the method of virtual noise.

We consider observations of a random field (or a random process), which is modeled by a nonlinear regression with a parametrized mean (or trend) and a parametrized covariance function. In the first part we show that under the assumption that the errors are normal with small variances, even when the number of observations is small, the ML estimators of both parameters are approximately unbiased, uncorrelated, with variances given by the inverse of the Fisher information matrix. In the second part we are extending the result of Pazman & Müller (2001) to the case of parametrized covariance function, namely we prove that the optimum designs with and without the presence of the virtual noise are identical. This in principle justify the use the method of virtual noise as a computational device also in this case. (authors' abstract)