Systematic Sampling with Unequal Probability and without Replacement
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不等概率不放回系统抽样
- 研究问题
- 按单位规模设置入样概率并采用随机起点系统抽样时，如何计算总体估计量的方差，排序又如何影响精度？
- 方法与思路
- 作者利用系统抽样的可实现样本结构，将估计量精确方差联系到特殊方差分析中的样本间均方，并提出两种近似方差估计。
- 材料陈述的贡献
- 摘要称，该程序结合简便抽样与精确方差公式，按规模排序时通常更有效率，但其方差估计依赖一定假设。
- 阅读关注点
- 应核对可实现样本数及其概率的推导、精确方差的适用条件、规模排序的作用，以及两种近似估计依赖的假设。
- 阅读边界
- 仅据所给 OpenAlex 索引摘要，稿件版本未注明；样本数、效率比较和方差估计均需核对具体设定，不能推广为普遍结论。
原文摘要
Given a population of N units, it is required to draw a sample of n distinct units in such a way that the probability for the i th unit to be in the sample is proportional to its ‘size’ x. From the alternative methods of achieving this we consider here only the so-called systematic method which, to the best of our knowledge, was first developed by W. G. Madow (1949): The units in the population are listed in a ‘particular’ order, their xi accumulated and a systematic selection of n elements from a ‘random start’ is then made on the accumulation. In a more recent paper (H. O. Hartley and J. N. K. Rao (1962)) an asymptotic estimation theory (for large N) associated with this procedure was developed for the case when the order of the listed units is random. In this paper we draw attention to certain properties of Madow's estimator: We utilize the fact that with systematic sampling the total number of different samples is N (rather than ( N n ) as with completely random sampling). This simplification in the definition of the variance of the estimator in repeated sampling enables us to identify the exact variance of Madow's estimator with a ‘between sample mean square’ in a special analysis of variance (see section 4) and compare it with the variance of the pps estimator in sampling with replacement as well as in other sampling procedures. We also develop two approximate methods of variance estimation (see section 5). We pay particular attention to the case when the units are listed in the order of their size. With this particular arrangement our method can be described as ‘systematic with random start’ and the gain in precision that we accomplish has of course, analogues in systematic sampling with equal probabilities employing ratio estimators in which there is a relation between the ratio ri = yi /xi and xi. Compared with other methods the present procedure combines the advantage of ease of systematic sample selection with the availability of exact variance formulas for any n and N. Moreover, it usually leads to a more efficient estimate. Its shortcoming resides in the fact that the estimation of the variance is based on certain assumptions.