On optimal variance estimation under different spatial subsampling schemes

This chapter provides 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, it is shown 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. With suitable functions, one can represent a wide range of estimators under the present framework. In particular, these include means, differences, and ratios of means, sample moments, spatial variograms and correlograms, likelihood-based estimators of process parameters, and test statistics for spatial autocorrelation. The extension of the consistency property of subsample variance estimation to more general spatial sampling and subsampling regions is also elaborated. 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.