The Importance of the Range Parameter for Estimation and Prediction in Geostatistics

Two canonical problems in geostatistics are estimating the parameters in a specified family of stochastic process models and predicting the process at new locations. A number of asymptotic results for these problems over a fixed spatial domain indicate that, for a Gaussian process with Mat\'ern covariance function, one can fix the range parameter controlling the rate of decay of the process and obtain results that are asymptotically equivalent to the case that the range parameter is known. We discuss why these results do not always provide the appropriate intuition for finite samples. Moreover, we prove that a number of these asymptotic results may be extended to the case that the variance and range parameters are jointly estimated via maximum likelihood or maximum tapered likelihood. Our simulation results show that performance on a variety of metrics is improved and asymptotic approximations are applicable for smaller sample sizes when the range parameter is estimated. These effects are particularly apparent when the process is mean square differentiable or the effective range of spatial correlation is small.