Variance Estimation for Statistics Computed from Spatial Lattice Data

SUMMARY A statistic s() is computed on spatially indexed data {X i: i ∈ D), where D is a finite subset of the integer lattice Z2. We propose a method for estimating the variance (and other moments) of s() by using only the data. The set D may be irregularly shaped, the statistic s() may be arbitrarily complicated and no distributional assumptions (marginal or joint) are necessary. The method uses the statistic computed on overlapping 'subshapes' of D as replicates of s(). The estimator is simply the sample variance of the (standardized) replicates. We demonstrate L 2-consistency of the estimator (under mild conditions on s() and the strength of spatial dependence). A rate of convergence is also given, giving guidance into the appropriate choice of subshape size, and the estimator is illustrated in two data examples.