Rejective sampling improves design and estimation efficiency of single-phase sampling when auxiliary information in a finite population is available. When such auxiliary information is unavailable, we propose to use two-phase rejective sampling (TPRS), which involves measuring auxiliary variables for the sample of units in the first phase, followed by the implementation of rejective sampling for the outcome in the second phase. We explore the asymptotic design properties of double expansion and regression estimators under TPRS. We show that TPRS enhances the efficiency of the double expansion estimator, rendering it comparable to a regression estimator. We further refine the design to accommodate varying importance of covariates and extend it to multi-phase sampling. We start with the theory for the population mean and then extend the theory to parameters defined by general estimating equations. Our asymptotic results for TPRS immediately cover the existing single-phase rejective sampling, under which the asymptotic theory has not been fully established.

翻译：当有限总体中存在辅助信息时，拒绝抽样能够提升单阶段抽样的设计与估计效率。若此类辅助信息不可得，本文提出采用两阶段拒绝抽样方法：第一阶段对样本单元测量辅助变量，第二阶段对结果变量实施拒绝抽样。我们研究了两阶段拒绝抽样框架下双重扩展估计量与回归估计量的渐近设计性质。研究表明，两阶段拒绝抽样能提升双重扩展估计量的效率，使其达到与回归估计量相当的水平。我们进一步优化设计方案以适应协变量的不同重要性，并将其扩展至多阶段抽样场景。本文首先建立总体均值的理论框架，随后将理论推广至由一般估计方程定义的参数。所建立的两阶段拒绝抽样渐近结果可直接涵盖现有单阶段拒绝抽样情形——该情形下的渐近理论此前尚未得到完整建立。