Minimax unbiased estimation for finite populations with bounded outcomes

2026-05-19

AI Summary

This study addresses the joint optimization of sampling design and estimation under bounded population values, aiming to achieve design-unbiased estimation of the population total while minimizing the worst-case mean squared error. Within the Horvitz–Thompson framework and adopting a minimax criterion, the paper establishes—for the first time—the minimax lower bound over all design-unbiased estimators and shows that this bound is attainable when the unit inclusion indicators are pairwise independent. The authors further propose a midpoint-differenced Horvitz–Thompson estimator, which achieves minimax optimality under an independent sampling strategy with inclusion probabilities πᵢ* = min(1, c(bᵢ − aᵢ)). This estimator is also shown to be admissible within the class of unbiased and affine-equivariant estimators, thereby extending Gabler’s (1990) linear result to a broader class of estimators.

Abstract

We study design-unbiased estimation of the finite-population total $\sum_{i=1}^N y_i$ when each outcome satisfies known bounds $y_i\in[a_i,b_i]$. For any sampling design with inclusion probabilities $π_i>0$, we prove a sharp lower bound on the worst-case squared error over the rectangular parameter space. This bound is attained if and only if the unit inclusion indicators are pairwise independent, in which case the minimax estimator is the midpoint-differenced Horvitz-Thompson estimator $\sum_{i=1}^N m_i+\sum_{i\in S}(y_i-m_i)/π_i$, with $m_i=(a_i+b_i)/{2}$. We then solve the joint design-and-estimation problem under the constraint $\sum_i π_i\le n$. We find that a minimax strategy samples units independently with probabilities $π_i^\ast=\min(1,c (b_i-a_i))$ where $c>0$ is chosen so that $\sum_i π_i^\ast=n$, and uses the midpoint-differenced estimator. This extends Gabler (1990)'s linear minimax result to the full class of design-unbiased estimators. We also show that the estimator is admissible among unbiased estimators and affine equivariant.

Problem

• minimax estimation
• finite population
• bounded outcomes
• design-unbiased estimation
• Horvitz-Thompson estimator

Innovation

• minimax estimation
• finite population
• design-unbiased estimator
• Horvitz-Thompson estimator
• pairwise independent sampling