Mean Squared Errors of Prediction by Kriging in Linear Models with Ar(1) Errors

Kriging, in the scientific literature, is used as a name for the theory of prediction in random processes (random fields) with an unknown mean value and, possibly, with an unknown covariance function. M. Stein in a series of articles (1988), (1990a), (1990b) and (1990c) studies the case when the unknown covariance function of the observed process is misspecified, but not estimated from the data. Limit theory of prediction of time series with estimated parameters has been studied by many authors including Bhansali (1981), Fuller and Hasza (1981), Kunimoto and Yamamoto (1985) and Toyooka (1982). Some of these authors have assumed that the mean value of the observed process is zero. Harville (1985), Harville and Jeske (1992) and Zimmerman and Cressie (1992) studied properties and approximations of the mean squared error of prediction with unbiasedly estimated parameters in the case when a covariance function depends linearly on unknown parameters. The main aim of this paper is to derive an approximate expression for the mean square error of a predictor with estimated parameters which is based on a finite observation of a stochastic process following a linear regression model with AR(1) errors. In this case the dependence of covariance function on unknown parameters is nonlinear.