This study considers the estimation of the direct bias-correction term for estimating the average treatment effect (ATE). Let $\{(X_i, D_i, Y_i)\}_{i=1}^{n}$ be the observations, where $X_i$ denotes $K$-dimensional covariates, $D_i \in \{0, 1\}$ denotes a binary treatment assignment indicator, and $Y_i$ denotes an outcome. In ATE estimation, $h_0(D_i, X_i) = \frac{1[D_i = 1]}{e_0(X_i)} - \frac{1[D_i = 0]}{1 - e_0(X_i)}$ is called the bias-correction term, where $e_0(X_i)$ is the propensity score. The bias-correction term is also referred to as the Riesz representer or clever covariates, depending on the literature, and plays an important role in construction of efficient ATE estimators. In this study, we propose estimating $h_0$ by directly minimizing the Bregman divergence between its model and $h_0$, which includes squared error and Kullback--Leibler divergence as special cases. Our proposed method is inspired by direct density ratio estimation methods and generalizes existing bias-correction term estimation methods, such as covariate balancing weights, Riesz regression, and nearest neighbor matching. Importantly, under specific choices of bias-correction term models and Bregman divergence, we can automatically ensure the covariate balancing property. Thus, our study provides a practical modeling and estimation approach through a generalization of existing methods.

翻译：本研究考虑估计平均处理效应（ATE）时直接偏误校正项的估计问题。令$\{(X_i, D_i, Y_i)\}_{i=1}^{n}$为观测数据，其中$X_i$表示$K$维协变量，$D_i \in \{0, 1\}$表示二元处理分配指标，$Y_i$表示结果变量。在ATE估计中，$h_0(D_i, X_i) = \frac{1[D_i = 1]}{e_0(X_i)} - \frac{1[D_i = 0]}{1 - e_0(X_i)}$被称为偏误校正项，其中$e_0(X_i)$为倾向得分。根据文献不同，该偏误校正项亦被称为Riesz表示子或巧妙协变量，在构建高效ATE估计量中具有重要作用。本研究提出通过直接最小化其模型与$h_0$之间的Bregman散度来估计$h_0$，该方法以平方误差和Kullback-Leibler散度作为特例。所提出的方法受直接密度比估计方法启发，并推广了现有的偏误校正项估计方法，如协变量平衡权重、Riesz回归和最近邻匹配。重要的是，在特定的偏误校正项模型和Bregman散度选择下，该方法能自动确保协变量平衡性质。因此，本研究通过对现有方法的推广，提供了一种实用的建模与估计框架。