AI 中文总结
该研究针对有限总体随机实验，将样本平均处理效应估计的极小极大风险归约为含2个未知参数的估计问题的风险，建立二阶风险展开式，证明标准程序仅一阶最优，给出达到极小极大风险的方法并讨论实际意义。
AI 中文摘要
我们研究有限总体随机实验中估计样本平均处理效应的极小极大最优设计与估计量，其中设计和估计量均无限制。对于二元潜在结果，我们证明该极小极大风险等价于含2个未知参数的估计问题的极小极大风险ρₙ^*。利用该归约，我们建立二阶风险展开式ρₙ^* = n⁻¹ - Cn⁻⁴/³ + oₙ(n⁻⁴/³)，其中显式常数C与Airy函数相关。极小极大风险可通过伯努利随机化结合非线性收缩估计量达到。我们的结果表明，完全随机化结合均值差这类标准程序仅在n的一阶项上达到极小极大最优。我们还推导了这些程序的容许性进一步结果，并讨论了研究结果的实际意义。
英文摘要
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.
Comments6 figures, 38 pages. Comments welcome
