Sampling with Varying Probabilities without Replacement: Rotating and Non-Rotating Samples

Abstract A population of N units is considered. It is assumed that each unit has a “size measure,” pi , associated with it (Σpi = 1). A sample of n units is to be selected one by one without replacement and in such a way that the probability for the i-th unit to be in the sample is npi. To achieve this a method is proposed whereby the probability for the i-th unit to be selected is equal to pi at each of the n successive selections. The procedure may be applied to one-stage sampling or to multi-stage sampling. In the latter case a self-weighting estimator is supplied, which is applicable even in the case where the sample at subsequent surveys is partially replaced (“rotating sample”).