目标极大似然估计的双稳健性及二阶偏差校正：一种半参数理论视角

Double Robustness and Second-Order Bias Correction of Targeted Maximum Likelihood Estimation: A Semiparametric Theoretical Perspective

摘要: 目标极大似然估计(Targeted Maximum Likelihood Estimation, TMLE)是一种在半参数和非参数模型中构造局部有效估计量的通用框架。本文从半参数理论出发，系统阐述TMLE的核心理论基础。首先，针对平均处理效应(ATE)参数，严格推导其有效影响函数(EIF)，并给出显式表达式。其次，深入分析TMLE的双重稳健性性质，通过偏差分解严格证明该性质成立的条件。作为重要参照，本文同时引入增强型逆概率加权估计量(AIPTW)，通过并列对比两者的估计方程与偏差结构，揭示TMLE在双重稳健性实现路径上的独特优势。第三，利用von Mises展开揭示TMLE将一阶偏差降为二阶偏差的机制，阐明二阶余项乘积结构保证偏差的快速收敛。第四，专门讨论TMLE的正则性条件，包括Donsker条件的统计学含义、n^−1/4收敛率要求的理论来源、交叉验证的绕行机制以及重叠性假设的影响。最后，证明TMLE的渐近正态性与半参数有效性。本文通过详细的数学推导，为研究者深入理解TMLE的半参数理论根基提供了系统性的技术基础。

Abstract: Targeted Maximum Likelihood Estimation (TMLE) is a general framework for constructing locally efficient estimators in semiparametric and nonparametric models. From a semiparametric perspective, this paper systematically presents the core theoretical foundations of TMLE. First, for the average treatment effect (ATE) parameter, we rigorously derive its efficient influence function (EIF) and provide an explicit expression. Second, we thoroughly analyze the double robustness property of TMLE and rigorously prove the conditions under which this property holds through bias decomposition. As an important reference, we simultaneously introduce the augmented inverse probability weighted estimator (AIPTW) and, by juxtaposing their estimating equations and bias structures, reveal the unique advantages of TMLE in achieving double robustness. Third, we employ the von Mises expansion to demonstrate how TMLE reduces the first‑order bias to second order, and we elucidate how the product structure of the second‑order remainder ensures fast convergence of the bias. Fourth, we specifically discuss the regularity conditions for TMLE, including the statistical meaning of the Donsker condition, the theoretical origin of the n^−1/4 convergence rate requirement, the mechanism by which cross‑fitting bypasses these conditions, and the impact of the overlap assumption. Finally, we prove the asymptotic normality and semiparametric efficiency of TMLE. Through detailed mathematical derivations, this paper provides a systematic technical foundation for researchers to gain a deep understanding of the semiparametric theoretical underpinnings of TMLE.