Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process

We consider an estimation problem with observations from a Gaussian process. The problem arises from a stochastic process modeling of computer experiments proposed recently by Sacks, Schiller, and Welch. By establishing various representations and approximations to the corresponding log-likelihood function, we show that the maximum likelihood estimator of the identifiable parameter [theta][sigma]2 is strongly consistent and converges weakly (when normalized by [radical sign]n) to a normal random variable, whose variance does not depend on the selection of sample points. Some extensions to regression models are also obtained.