We consider the problem of causal inference based on observational data (or the related missing data problem) with a binary or discrete treatment variable. In that context we study counterfactual density estimation, which provides more nuanced information than counterfactual mean estimation (i.e., the average treatment effect). We impose the shape-constraint of log-concavity (a unimodality constraint) on the counterfactual densities, and then develop doubly robust estimators of the log-concave counterfactual density (based on an augmented inverse-probability weighted pseudo-outcome), and show the consistency in various global metrics of that estimator. Based on that estimator we also develop asymptotically valid pointwise confidence intervals for the counterfactual density.

翻译：我们考虑基于观测数据（或相关的缺失数据问题）进行因果推断的问题，其中涉及二元或离散处理变量。在此背景下，我们研究反事实密度估计，该估计比反事实均值估计（即平均处理效应）能提供更细致的信息。我们对反事实密度施加对数凹性（一种单峰性约束）形状约束，然后开发出基于增强逆概率加权伪结果的对数凹反事实密度的双重稳健估计量，并证明该估计量在各种全局度量下的一致性。基于该估计量，我们还构建了反事实密度的渐近有效点态置信区间。