Plane sampling
Consider a population of \(N=N_ 1\cdot N_ 2\) units arranged in a grid of \(N_ 1=n_ 1\cdot k_ 1\) rows and \(N_ 2=n_ 2\cdot k_ 2\) columns. The population mean is estimated by the sample mean and the efficiency of various sample designs is compared when the population size N and the sample size n tend to infinity in such a way that the ratio N/n is constant. Samples aligned in both directions and unaligned samples are studied. The population values are assumed to arise as a realization of a spatial process, stationary in the wide sense with a non-negative autocorrelation function \(\rho\) being non-increasing in both arguments. Assuming that the correlation function \(\rho\) satisfies \[ \sum_{u}\sum_{v}\rho (| u|,| v|)<\infty \] it is shown, both for unaligned and aligned designs, that systematic sampling is at least as good as stratified sampling which is at least as good as simple random sampling. Under an additional assumption about weak dependence of the spatial process generating the population, asymptotic normality of the aligned systematic sample mean is shown.
scientific article; zbMATH DE number 3951779
scientific article