Computing inclusion probabilities to obtain Horvitz–Thompson estimators for sampling plans excluding neighbouring units

Abstract

Following the relative efficiency comparison of sampling plans excluding certain neighbouring units [See, K., Noble, R.B. and Bailer, J., 2005, Comparisons of relative efficiencies of sampling plans excluding certain neighboring units: a simulation study. Journal of Statistical Computation and Simulation, in press.], we now compute the inclusion probabilities for Horvitz–Thompson estimators (HTE) of the population mean (or total) and their variances, for various sampling plans that exclude certain neighbouring units (SENU) [See, K., Stufken, J., Song, S.Y. and Bailer, A.J., 2000, Relative efficiencies of sampling plans for selecting a small number of units from a rectangular region. Journal of Statistical Computation and Simulation, 66(4), 273–294.]. Sampling plans excluding neighbouring units type sampling plans can be thought of as two-dimensional cases of balanced sampling excluding contiguous units (BSEC) [Hedayat, A.S., Rao, C.R. and Stufken, J., 1988, Sampling designs excluding contiguous units. Journal of Statistical Planning and Inference, 19, 159–170.]. For the SENU-type sampling plans we consider, we provide a strategy for estimating the first-order and second-order inclusion probabilities. These probabilities are critical components for constructing the HTE of the mean or total and their variances. For practical sample sizes, these calculations can be based on full enumerations of the design space, whereas larger population grids and sample sizes may require the sampling of the design space to estimate these probabilities. Illustrations are provided for 5×5 and 10×10 grids. For the 5×5 grid, an exact solution is obtained from an exhaustive search for the first- and second-order probabilities, and therefore, the HTE and their variances can be complete. For the 10×10 grid, we obtain the inclusion probabilities based on the simulations, and thus, one can estimate the HTE and their variances.