  We consider estimation of the parameters of a Gaussian Stochastic Process
(GaSP), in the context of emulation (approximation) of computer models for
which the outcomes are real-valued scalars. The main focus is on estimation of
the GaSP parameters through various generalized maximum likelihood methods,
mostly involving finding posterior modes; this is because full Bayesian
analysis in computer model emulation is typically prohibitively expensive. The
posterior modes that are studied arise from objective priors, such as the
reference prior. These priors have been studied in the literature for the
situation of an isotropic covariance function or under the assumption of
separability in the design of inputs for model runs used in the GaSP
construction. In this paper, we consider more general designs (e.g., a Latin
Hypercube Design) with a class of commonly used anisotropic correlation
functions, which can be written as a product of isotropic correlation
functions, each having an unknown range parameter and a fixed roughness
parameter. We discuss properties of the objective priors and marginal
likelihoods for the parameters of the GaSP and establish the posterior
propriety of the GaSP parameters, but our main focus is to demonstrate that
certain parameterizations result in more robust estimation of the GaSP
parameters than others, and that some parameterizations that are in common use
should clearly be avoided. These results are applicable to many frequently used
covariance functions, e.g., power exponential, Mat{\'e}rn, rational quadratic
and spherical covariance. We also generalize the results to the GaSP model with
a nugget parameter. Both theoretical and numerical evidence is presented
concerning the performance of the studied procedures.
