简单随机抽样中的“虚拟普查”技术

“Imaginary Census” in Simple Random Sampling

简单随机抽样,包括有无放回两种形式,是最基础的抽样设计.本文基于“虚拟普查”想法,对简单估计量的抽样方差给出了新的计算思路,并揭示了有无放回简单随机抽样之间的内在联系.“虚拟普查”想法的核心在于虚拟普查矩阵.该矩阵记录了逐一抽取的无放回简单随机抽样,假设拓展为普查时--“虚拟普查”名称的由来,所有总体单元的入样轨迹.论文构建的“虚拟普查”技术框架,可以看作已有对称化论证方法和以入样指示变量为工具的论证方法的融合,对增进理解传统抽样策略,看起来具有潜在价值.作为示例,论文针对实践中常用的两个抽样策略,有放回不等概抽样和自适应整群抽样,给出了基于简单随机抽样的解读视角.

Simple random sampling,both with and without replacement,are fundamental in traditional survey sampling.Based on an idea of“imaginary census”,we provide in this note a new way for calculating the sample variance of simple sample average,as well as understanding the intrinsic relationship between simple random sampling with and without replacement.The key concept is an“imaginary census”matrix,which records the exact sampling trajectory of each draw without replacement,until all population units were sampled out.The random matrix possesses a nice probabilistic symmetry,and each of its column summed to be a fixed number.The new framework,in a sense,is a fusion of two existing classic techniques in traditional survey sampling.One depends on the random vector of 0–1 valued random variables indicating which population units were sampled,and the other is the symmetrization technique.Our method appears valuable for understanding several important sampling strategies,even to the branch of survey sampling itself.For illustration,we present two examples pertain to unequal probability sampling with replacement and adaptive cluster sampling,with a focus on understanding these sampling strategies from the perspective of simple random sampling.