Conditional Optimal Sampling: Enhancing Unequal-Probability Designs with Auxiliary Information

Abstract

We address the problem of selecting a fixed-size sample from a finite population when the goal is to reduce the estimation error as much as possible. Our approach builds on the idea of choosing samples that are nearly optimal with respect to the values of the study variable, even though these values are not observed in advance. Instead, we rely on predictions derived from auxiliary information, for example through statistical or machine-learning models, and construct sampling designs that are optimal for these predicted values. We show that this strategy is remarkably effective: even when the predictions are imperfect, the resulting designs tend to be very close to the unknown optimal design based on the true values. To understand why this happens, we study the geometric structure underlying the space of all feasible sampling designs and use this perspective to develop practical algorithms. Simulation experiments indicate that the proposed methodology achieves noticeably lower estimation error than classical unequal-probability sampling methods commonly used in the literature, including balanced procedures such as cube and classical fixed-size designs like conditional Poisson sampling.