This study designs an adaptive experiment for efficiently estimating average treatment effect (ATEs). We consider an adaptive experiment where an experimenter sequentially samples an experimental unit from a covariate density decided by the experimenter and assigns a treatment. After assigning a treatment, the experimenter observes the corresponding outcome immediately. At the end of the experiment, the experimenter estimates an ATE using gathered samples. The objective of the experimenter is to estimate the ATE with a smaller asymptotic variance. Existing studies have designed experiments that adaptively optimize the propensity score (treatment-assignment probability). As a generalization of such an approach, we propose a framework under which an experimenter optimizes the covariate density, as well as the propensity score, and find that optimizing both covariate density and propensity score reduces the asymptotic variance more than optimizing only the propensity score. Based on this idea, in each round of our experiment, the experimenter optimizes the covariate density and propensity score based on past observations. To design an adaptive experiment, we first derive the efficient covariate density and propensity score that minimizes the semiparametric efficiency bound, a lower bound for the asymptotic variance given a fixed covariate density and a fixed propensity score. Next, we design an adaptive experiment using the efficient covariate density and propensity score sequentially estimated during the experiment. Lastly, we propose an ATE estimator whose asymptotic variance aligns with the minimized semiparametric efficiency bound.

翻译：本研究设计了一种自适应实验，旨在高效估计平均处理效应（ATE）。我们考虑一个自适应实验场景，其中实验者根据其决定的协变量密度依次采样实验单元并分配处理。分配处理后，实验者立即观测相应结果。实验结束时，实验者利用收集的样本估计ATE。实验者的目标是以更小的渐近方差估计ATE。现有研究已设计出自适应优化倾向得分（处理分配概率）的实验。作为此类方法的推广，我们提出一个框架，使实验者能够同时优化协变量密度与倾向得分，并发现同时优化两者比仅优化倾向得分更能降低渐近方差。基于此思路，在实验的每一轮中，实验者根据历史观测优化协变量密度与倾向得分。为设计自适应实验，我们首先推导出最小化半参数效率界（在固定协变量密度与倾向得分下渐近方差的下界）的有效协变量密度与倾向得分。其次，利用实验过程中序贯估计的有效协变量密度与倾向得分设计自适应实验。最后，我们提出一个ATE估计量，其渐近方差与最小化的半参数效率界一致。