发表机构
The University of British Columbia（不列颠哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明，与SRSWOR具有相同一阶和二阶包含概率的抽样设计仍可能导致样本总和收敛于非正态分布，表明低阶包含概率不足以决定渐近正态性。
AI 中文摘要
调查抽样中的中心极限定理通常是在特定抽样设计下建立的，而非基于设计的通用特征。受寻找更一般条件的动机驱动，我们研究低阶包含概率能否提供这样的基础。一阶和二阶包含概率决定了抽样估计量（如霍维茨-汤普森估计量）的前两阶矩，但我们证明，即使这些概率与无放回简单随机抽样（SRSWOR）的完全一致，也不能保持渐近正态性。我们构造了一系列固定规模抽样设计，其与SRSWOR具有完全相同的一阶和二阶包含概率，同时构造了一系列有限总体，在这些总体下，样本总和在SRSWOR下收敛于N(0,1)，而在替代设计下收敛于中心化的标准指数分布。因此，两个样本总和在序列的每个阶段都具有完全相同的均值和方差，却具有不同的极限分布。替代设计由有限射影平面产生的平衡不完全区组设计获得。其循环结构使我们能够规定区组抽样下样本总和的分布，而概率置换论证结合无放回抽样的指数不等式，产生了满足SRSWOR中心极限定理的有限总体。结果表明，低阶包含概率不能控制与渐近正态性相关的全局依赖结构，因此它们本身不能为复杂抽样设计下的中心极限定理提供一般条件。
英文摘要
Central limit theorems in survey sampling are typically established under specific sampling designs rather than from generic characteristics of a design. Motivated by the search for more general conditions, we investigate whether low-order inclusion probabilities can provide such a basis. First- and second-order inclusion probabilities determine the first two moments of sampling estimators such as the Horvitz--Thompson estimator, but we show that even exact agreement of these probabilities with those of simple random sampling without replacement (SRSWOR) does not preserve asymptotic normality. We construct a sequence of fixed-size sampling designs having exactly the same first- and second-order inclusion probabilities as SRSWOR, together with a sequence of finite populations for which the sample total converges to $N(0,1)$ under SRSWOR but to a centered standard exponential distribution under the alternative designs. Thus, the two sample totals have exactly the same mean and variance at every stage of the sequence, yet have different limiting distributions. The alternative designs are obtained from balanced incomplete block designs arising from finite projective planes. Their cyclic structure allows us to prescribe the distribution of the sample total under block sampling, while a probabilistic permutation argument, combined with an exponential inequality for sampling without replacement, yields finite populations satisfying the SRSWOR central limit theorem. The result shows that low-order inclusion probabilities do not control the global dependence structure relevant to asymptotic normality and therefore cannot, by themselves, provide general conditions for central limit theorems under complex sampling designs.
Comments15 pages, 1 figure
