With‐Replacement Balanced Sampling

ABSTRACT Balanced sampling ensures that selected samples allow Horvitz–Thompson estimators to reproduce known auxiliary totals. However, no efficient procedures currently exist for producing balanced samples with replacement despite their potential applications to balanced imputations or balanced stream sampling. This article introduces three new with‐replacement balanced sampling algorithms, built on a generalization of the cube method, and proposes a variance approximation tailored to such designs. A simulation study evaluates the accuracy of this approximation under both idealized conditional multinomial sampling and the proposed algorithms. We also demonstrate how these methods can be used to construct a single‐pass balanced stream sampling procedure, which maintains balance on auxiliary variables while ensuring expected sample counts proportional to size.