Asymptotically Efficient Prediction of a Random Field with a Misspecified Covariance Function

Abstract

Best linear unbiased predictors of a random field can be obtained if the covariance function of the random field is specified correctly. Consider a random field defined on a bounded region $R$. We wish to predict the random field $z(\cdot)$ at a point $x$ in $R$ based on observations $z(x_1), z(x_2), \ldots, z(x_N)$ in $R$, where $\{x_i\}^\infty_{i = 1}$ has $x$ as a limit point but does not contain $x$. Suppose the covariance function is misspecified, but has an equivalent (mutually absolutely continuous) corresponding Gaussian measure to the true covariance function. Then the predictor of $z(x)$ based on $z(x_1), \ldots, z(x_N)$ will be asymptotically efficient as $N$ tends to infinity.