Sharp Minimax Theory for Randomized Experiments

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 ${\rho }_n^{\ast }$ of an estimation problem with $2$ unknown parameters. We leverage this reduction to establish a second-order risk expansion ${\rho }_n^{\ast }=n^{-1}-C{n}^{-4\mathrm{/}3}+o_n(n^{-4\mathrm{/}3})$ 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 $n\mathrm{.}$ We derive further results on admissibility of these procedures and discuss the practical implications of our results.