Multistage sampling with partial replacement

Multistage sampling is a cost-effective technique for obtaining samples in large-scale sampling situations. If the sampling is repeated over time, the application of successive sampling with partial replacement (SPR) is advantageous. In this paper, some theory of SPR when the sampling design is multistage, is considered. Estimators of the population mean and the change in the mean over successive occasions, together with their respective variances, are derived without the restrictive assumptions of equal sample sizes and (or) equal variances at each stage of the multistage design on the successive occasions. The derivation is restricted to a two-stage design on two successive occasions, although sufficient background is presented to enable the reader to extend the theory to other than the two-stage design.