Subsampling a Random Field

Subsampling techniques developed by Hartigan (for independent observations) and by Carlstein (for mixing processes on the integers) are extended to estimate variances and covariances of statistics of spatial processes. If the process is stationary, and dependence weakens rapidly with increasing distance, then the procedure is consistent in the sense that, as the region available for observation grows large, both the bias and the variance, of the estimator of the second moments, converge to zero. The technique is applied to estimate the variance of the sample intensity of a binary Markov random field, and the variance of an index of clumping for spatial point processes studied by quadrat methods. Antonio Possolo is Assistant Professor, Department of Statistics GN-22, University of Washington, Seattle, WA 98195. This work was carried out under Office of Naval Research Contract NOOOI4-85-K-0422. Edward Carlstein (University of North Carolina) kindly made his doctoral dissertation available to the author prior to publication. AMS 1980 subject classification. Primary 62M05; Secondary 62M30, 60060, 60055.