We study the problem of estimating the average treatment effect (ATE) under sequentially adaptive treatment assignment mechanisms. In contrast to classical completely randomized designs, we consider a setting in which the probability of assigning treatment to each experimental unit may depend on prior assignments and observed outcomes. Within the potential outcomes framework, we propose and analyze two natural estimators for the ATE: the inverse propensity weighted (IPW) estimator and an augmented IPW (AIPW) estimator. The cornerstone of our analysis is the concept of design stability, which requires that as the number of units grows, either the assignment probabilities converge, or sample averages of the inverse propensity scores and of the inverse complement propensity scores converge in probability to fixed, non-random limits. Our main results establish central limit theorems for both the IPW and AIPW estimators under design stability and provide explicit expressions for their asymptotic variances. We further propose estimators for these variances, enabling the construction of asymptotically valid confidence intervals. Finally, we illustrate our theoretical results in the context of Wei's adaptive coin design and Efron's biased coin design, highlighting the applicability of the proposed methods to sequential experimentation with adaptive randomization.

翻译：我们研究了在序列自适应处理分配机制下估计平均处理效应（ATE）的问题。与经典的完全随机设计不同，我们考虑了一种设置，其中为每个实验单元分配处理的概率可能依赖于先前的分配和观测到的结果。在潜在结果框架内，我们提出并分析了两种针对ATE的自然估计量：逆倾向加权（IPW）估计量和一个增广IPW（AIPW）估计量。我们分析的基石是设计稳定性的概念，它要求随着单元数量的增长，要么分配概率收敛，要么逆倾向得分及逆补倾向得分的样本平均值依概率收敛到固定的、非随机的极限。我们的主要结果在设计稳定性条件下为IPW和AIPW估计量建立了中心极限定理，并给出了它们渐近方差的显式表达式。我们进一步提出了这些方差的估计量，从而能够构建渐近有效的置信区间。最后，我们在Wei的自适应硬币设计和Efron的有偏硬币设计的背景下阐述了我们的理论结果，突显了所提方法在采用自适应随机化的序列实验中的适用性。