On the Selection of $n$ Primary Sampling Units From a Stratum Structure $(n \geqq 2)$

Abstract

Hansen and Hurwitz [1] showed for two-stage sampling that the selection of a single primary sampling unit (p.s.u.) with probability proportional to some measure of its size (p.p.s.) from each stratum is generally more efficient than selection with equal probability. Midzuno [5] generalised the Hansen and Hurwitz approach to sampling a combination of $n$ elements from each stratum. Neither Hansen and Hurwitz nor Midzuno provided a method for estimating the between component of the total error from the sample. Recently Horvitz and Thomson [3] have also given a method for dealing with sampling without replacement when arbitrary probabilities of selection are used for elements remaining prior to each draw. Methods for obtaining an unbiased estimate of the population total as well as of the variance of the estimate are presented. The scheme, however, suffers from certain practical disadvantages. One such disadvantage is the difficulty involved in the determination of the selection probabilities. Another disadvantage is that Horvitz and Thomson's unbiased estimate of the variance has generally no practical application as it may assume negative values. This has been shown independently by the present author [9], [10] and Yates and Grundy [11]. The authors also derived an expression of the unbiased estimate which is free from this defect. Working independently, the present author [7], [8] developed the theory when a combination of $n$ p.s.u.'s are sampled from a stratum and applied it to the case when $n = 2$. In this paper an outline is given of the general theory of the selection and estimation procedure for obtaining unbiased estimates of the between component of total error where first $r$ p.s.u.'s are selected with p.p.s. and the remaining $n - r$ are selected with equal probability, the selection being without replacement. An expression for the estimate of the variance of the estimated total is presented. It is shown that the unbiased estimate of the variance of the estimate is generally inefficient and may assume negative values for certain combinations of the sample values except for the special case when the measures of sizes are all equal. A biased estimate has, however, been derived which is always positive and is more efficient than the unbiased estimate. It is shown that for the particular case when $r = 1$ the unbiased estimate of the total reduces to a simple form which is useful in practice. It is proved that the selection of one p.s.u. with p.p.s. and the remaining $n - 1$ with equal probability is equivalent to selecting a combination of $n$ p.s.u.'s, where the measure of size is the sum of the measures of the combination.