Estimating the prediction mean squared error in Gaussian stochastic processes with exponential correlation structure

Given one or more realizations from the finite dimensional marginal distrib ution of a stochastic process, we consider the problem of estimating the sq uared prediction error when predicting the process at unobserved locations. An approximation taking into account the additional variability due to est imating parameters involved in the correlation structure was developed by K ackar & Harville (1984) and was revisited by Harville & Jeske (1992) as wel l as Zimmerman & Cressie (1992). The present paper discusses an extension o f these methods. The approaches will be compared via an extensive simulatio n study for models with and without random error term. Effects due to the d esigns used for prediction and for model fitting as well as due to the stre ngth of the correlation between neighbouring observations of the stochastic process are investigated. The results show that considering the additional variability in the predictor due to estimating the covariance structure Is of great importance and should not be neglected in practical applications.