Fixed-domain asymptotics of the maximum likelihood estimator and the Gaussian process approach for deterministic models

The fixed-domain asymptotics of the maximum likelihood estimator is studied in the framework of the Gaussian process approach for data collected as precise observations of a deterministic computer model given by an analytic function. It is shown that the maximum likelihood estimator of the correlation parameter of a Gaussian process does not converge to a finite value and the computational stability strongly depends on the type of the correlation function. In particular, computations are the most unstable for the Gaussian correlation function, which is typically used in the analysis of computer experiments, and significantly less unstable for the stable correlation function ρ(t)=e−∣t∣γ even if γ=1.9γ=1.9 which is close to 2.