In the first chapter we review some standard estimators in sampling from a finite population, and some design-based estimators in sampling from a continuous universe.
In concert with the theory initiated by professor Douglas Robson (personal communication) and later presented by Cordy (1993), we consider design-based variance estimation for probability sampling from a continuous and spatially distributed universe. Using this theory in chapter two, the sampling design of one random point from each cell of a translated grid is investigated and the problem of edge effects on estimation is illustrated with examples. Also in chapter four, standard systematic sampling methods from a finite population are reviewed. Then, for systematic samples drawn from a continuous universe, a new approach for investigating the estimators of the parameters of interest is introduced. This new approach can be useful for deriving unbiased variance estimators for many spatial systematic sampling methods and allows for proposing new efficient systematic sampling designs. For these systematic sampling designs, we present the estimator of the population total and the estimator of the variance for a population with one dimension, and we derive in general these estimators for n-dimensional population. Furthermore in chapter five, a mean-balanced sample of size two from each cell of a translated grid is investigated. Then an unbiased estimator of the population total is presented. Also, explicit formulas for the inclusion density functions are derived
