On optimal variance estimation under different spatial subsampling schemes

We provide a comprehensive examination of subsampling methods for spatial data on a grid. The considered subsamples may be scaled-down copies of the original sampling region or have a freely chosen shape. We derive the mean square error associated with general subsampling methods for estimating the variance of a large class of estimators. This yields an expression for the optimal subsample size for a given subsample shape. However, in contrast to the time series case, we show that the optimal subsample size and performance with each spatial subsampling method depends on the geometry of the sampling and subsampling regions in a nontrivial way. Examples for a few simple cases are presented to illustrate that both subsample size and shape may be selected to optimize spatial subsampling for variance estimation.