Sharp Minimax Theory for Randomized Experiments

2026-08-13

AI Summary

This study addresses the minimax optimal design problem for estimating the sample average treatment effect with binary outcomes in finite-population randomized experiments. By reformulating risk equivalence as a two-parameter estimation task, we derive an exact second-order asymptotic expansion involving Airy functions. We propose a novel framework combining Bernoulli randomization with nonlinear shrinkage estimation, obtaining explicit second-order risk constants and establishing its minimax optimality. Our results demonstrate that while the traditional difference-in-means estimator achieves only first-order optimality, the proposed estimator significantly outperforms standard procedures at the second order. These findings elucidate the limitations of existing methods and underscore the practical significance of second-order refinements in experimental design.

Abstract

We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $ρ_n^*$ of an estimation problem with $2$ unknown parameters. We leverage this reduction to establish a second-order risk expansion $ρ_n^* = n^{-1} - Cn^{-4/3} + o_n(n^{-4/3})$ for an explicit constant $C$ related to the Airy function. The minimax risk is attained by Bernoulli randomization with a nonlinear shrinkage estimator. Our results show that standard procedures such as complete randomization with difference in means are only minimax optimal up to first order in $n.$ We derive further results on admissibility of these procedures and discuss the practical implications of our results.

Problem

• Minimax Theory
• Randomized Experiments
• Sample Average Treatment Effect
• Finite Population
• Optimal Design

Innovation

• Minimax Theory
• Randomized Experiments
• Second-order Risk Expansion
• Nonlinear Shrinkage Estimator
• Bernoulli Randomization