Doubly robust estimation and inference for a log-concave counterfactual density

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.