A New Central Limit Theorem for the Augmented IPW Estimator: Variance Inflation, Cross-Fit Covariance and Beyond
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AI 根据所提供的论文材料归类：摘要直接研究高维协变量下平均处理效应的交叉拟合 AIPW 推断。
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增广逆概率加权估计量的新中心极限定理：方差膨胀与交叉拟合协方差
- 研究问题
- 当协变量维度与样本量同阶且参数不稀疏时，交叉拟合 AIPW 平均处理效应估计量的极限分布如何偏离经典结论？
- 方法与思路
- 作者在结果回归与倾向得分模型正确设定及协变量分布条件下，结合近似消息传递、确定性等价和留一法推导中心极限定理。
- 材料陈述的贡献
- 摘要称理论量化了高维下的方差膨胀，并揭示参与交叉拟合的估计量间协方差在根号样本量尺度下仍不可忽略。
- 阅读关注点
- 应核对协变量分布、模型设定与维度比例的条件，以及极限方差如何计算、交叉拟合如何划分和协方差项如何进入推断。
- 阅读边界
- 这是 arXiv 预印本，不能据此认定已同行评审或期刊接收；摘要未给出方差公式及完整条件，模拟稳健性不能替代模型误设理论。
原文摘要
Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.