Weighting methods are essential tools for estimating causal effects in observational studies, with the goal of balancing pre-treatment covariates across treatment groups. Traditional approaches pursue this objective indirectly, for example, via inverse propensity score weighting or by matching a finite number of covariate moments, and therefore do not guarantee balance of the full joint covariate distributions. Recently, distributional balancing methods have emerged as robust, nonparametric alternatives that directly target alignment of entire covariate distributions, but they lack a unified framework, formal theoretical guarantees, and valid inferential procedures. We introduce a unified framework for nonparametric distributional balancing based on the characteristic function distance (CFD) and show that widely used discrepancy measures, including the maximum mean discrepancy and energy distance, arise as special cases. Our theoretical analysis establishes conditions under which the resulting CFD-based weighting estimator achieves $\sqrt{n}$-consistency. Since the standard bootstrap may fail for this estimator, we propose subsampling as a valid alternative for inference. We further extend our approach to an instrumental variable setting to address potential unmeasured confounding. Finally, we evaluate the performance of our method through simulation studies and a real-world application, where the proposed estimator performs well and exhibits results consistent with our theoretical predictions.

翻译：加权方法是观测性研究中估计因果效应的关键工具，其目标在于平衡处理组间的预处理协变量。传统方法间接追求这一目标，例如通过逆倾向得分加权或匹配有限数量的协变量矩，因此无法保证完整联合协变量分布的平衡。近年来，分布平衡方法作为稳健的非参数替代方案出现，直接针对整个协变量分布的对齐，但它们缺乏统一框架、正式的理论保证以及有效的推断程序。我们提出了一个基于特征函数距离（CFD）的非参数分布平衡统一框架，并证明包括最大均值差异和能量距离在内的常用差异度量均可作为其特例。我们的理论分析建立了基于CFD的加权估计量达到$\sqrt{n}$相合性的条件。由于标准自助法可能对此估计量失效，我们提出子抽样作为有效的推断替代方案。我们进一步将方法扩展至工具变量设定，以处理潜在的未测量混杂。最后，我们通过模拟研究和实际应用评估了所提方法的性能，结果表明该估计量表现良好，且结果与我们的理论预测一致。