Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics

It is shown that in model-based geostatistics, not all parameters in the Matérn class can be estimated consistently if data are observed in an increasing density in a Ž xed domain, regardless of the estimation methods used. Nevertheless, one quantity can be estimated consistently by the maximum likelihoodmethod, and this quantity is more important to spatial interpolation. The results are established by using the properties of equivalence and orthogonality of probability measures. Some sufŽ cient conditions are provided for both Gaussian and non-Gaussian equivalent measures, and necessary conditions are provided for Gaussian equivalent measures. Two simulation studies are presented that show that the Ž xed-domain asymptotic properties can explain some Ž nite-sample behavior of both interpolation and estimation when the sample size is moderately large.