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.