Comparing counterfactual distributions can provide more nuanced and valuable measures for causal effects, going beyond typical summary statistics such as averages. In this work, we consider characterizing causal effects via distributional distances, focusing on two kinds of target parameters. The first is the counterfactual outcome density. We propose a doubly robust-style estimator for the counterfactual density and study its rates of convergence and limiting distributions. We analyze asymptotic upper bounds on the $L_q$ and the integrated $L_q$ risks of the proposed estimator, and propose a bootstrap-based confidence band. The second is a novel distributional causal effect defined by the $L_1$ distance between different counterfactual distributions. We study three approaches for estimating the proposed distributional effect: smoothing the counterfactual density, smoothing the $L_1$ distance, and imposing a margin condition. For each approach, we analyze asymptotic properties and error bounds of the proposed estimator, and discuss potential advantages and disadvantages. We go on to present a bootstrap approach for obtaining confidence intervals, and propose a test of no distributional effect. We conclude with a numerical illustration and a real-world example.

翻译：比较反事实分布能够为因果效应提供比典型汇总统计量（如均值）更为细致且更有价值的度量。本文考虑通过分布距离来刻画因果效应，重点关注两类目标参数。第一类是反事实结果密度。我们提出了一种双重稳健式估计量用于反事实密度，并研究了其收敛速率与极限分布。我们分析了所提估计量在$L_q$风险及积分$L_q$风险上的渐近上界，并提出了一种基于自助法的置信带。第二类是由不同反事实分布间的$L_1$距离定义的一种新颖的分布因果效应。我们研究了估计该分布效应的三种方法：对反事实密度进行平滑、对$L_1$距离进行平滑，以及施加边界条件。针对每种方法，我们分析了所提估计量的渐近性质与误差界，并讨论了其潜在优势与局限。进一步，我们提出了一种基于自助法构建置信区间的方法，并设计了一种无分布效应的检验。最后，我们通过数值模拟与真实案例进行了说明。