Optimal Strategies in 2-Stage Sampling

Abstract

In many survey situations simple random sampling of units and estimation of a total of interest by the expansion estimator are attractive methods, at least at first sight. Considering cost aspects suggests rather to use multiple stage sampling which, in general, is cheaper, but less effective. The design effect is an adequate criterion of the decrease of efficiency. We discuss this criterion for clusters (primary units) of equal size and derive exact conditions for a decrease of efficiency. The equality condition for cluster sizes seems not to be very restrictive, because in many cases one will be interested in clusters of approximately the same size, or, if sizes differ essentially, the clusters are partitioned into strata according to their sizes and the procedures for different strata are independent, each dealing with clusters of equal size or nearly so. In the context considered the use of the Horvitz–Thompson estimator is quite general. We examine a class of estimators with the Horvitz–Thompson estimator and a straight forward modification of it as special elements. As for the design effect all elements of the class are very similar, as for other aspects such as admissibility there are remarkable differences.