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 inference for the counterfactual density functions and contrasts thereof, which can provide more nuanced information than counterfactual means and the average treatment effect. We impose the shape-constraint of log-concavity, a type of unimodality constraint, on the counterfactual densities, and then develop doubly robust estimators of the log-concave counterfactual density based on augmented inverse-probability weighted pseudo-outcomes. We provide conditions under which the estimator is consistent in various global metrics. We also develop asymptotically valid pointwise confidence intervals for the counterfactual density functions and differences and ratios thereof, which serve as a building block for more comprehensive analyses of distributional differences. We also present a method for using our estimator to implement density confidence bands.

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，其中处理变量为二元或离散变量。在此背景下，我们研究反事实密度函数及其对比的推断，这能提供比反事实均值与平均处理效应更细致的信息。我们对反事实密度施加对数凹性这一形状约束（一种单峰性约束），进而基于增广逆概率加权伪结果构建对数凹反事实密度的双重稳健估计量。我们给出了该估计量在多种全局度量下具有一致性的条件。我们还为反事实密度函数及其差异与比值构建了渐近有效的逐点置信区间，这为更全面的分布差异分析奠定了基础。此外，我们提出了一种利用该估计量实现密度置信带的方法。