Estimation of random fields by piecewise constant estimators

The problems of designing the efficient sampling designs for estimation of random fields by piecewise constant estimators are studied, which is done asymptotically, namely, as the sample size goes to infinity. The performance of sampling designs is measured by the integrated mean-square error. Here, the sampling domain is properly partitioned into a number of subregions, and each subregion is further tessellated into regular diamonds when the covariance is a function of L1 norm, or regular hexagons if it is a function of L2 norm. The sizes of the regular diamonds or hexagons are determined by a density function. It turns out that if the density function is properly chosen, the centers of these diamonds or hexagons, as sampling points, are asymptotically optimal. Examples with Gaussian, a distorted Ornstein-Uhlenbeck and a non-product-type covariance are considered.