Balanced Sampling With Inequalities: Application to Category Bounding, Matrix Rounding, and Spread Sampling

In this article, we propose a novel algorithm for balanced sample selection with linear inequality constraints, ensuring that estimators remain within fixed bounds. This algorithm extends the cube method of Deville and Tillé, allowing the selection of a sample from a database where Horvitz-Thompson estimators of totals are equal or nearly equal to the true population totals. The new algorithm has several key applications, including imposing minimum sample sizes for small areas and constraining sample sizes in potentially overlapping categories. It also addresses the controlled rounding matrix problem and links to systematic sampling with unequal probabilities. It can also be used to select doubly stratified samples when the sums of the inclusion probabilities in the strata are not integer. Additionally, the algorithm enables the selection of spatially spread samples. Simulations demonstrate that this new method performs comparably to other spread sampling techniques. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

本文提出一种带有线性不等式约束的均衡样本选择新算法，可确保估计量始终落在固定区间内。该算法拓展了Deville与Tillé提出的立方抽样法（cube method），可从数据库中抽取样本，使得总体总量的霍维茨-汤普森估计量（Horvitz-Thompson estimator）与真实总体总量相等或近似相等。该新算法具备多项核心应用场景，包括为小区域设定最小样本量、对潜在重叠分类中的样本量进行约束；此外还可解决受控舍入矩阵（controlled rounding matrix）问题，并与不等概率系统抽样建立关联。当各层的入样概率之和非整数时，该算法还可用于选取双重分层样本。此外，该算法能够实现空间分散样本的选取。仿真实验表明，该新方法的表现与其他分散抽样技术相当。本文的补充材料可在线获取，其中包含可用于复现研究成果的材料的标准化说明。