A doubly robust weighting estimator of the average treatment effect on the treated

We introduce an importance sampling derivation of the average treatment effect on the treated and extend this to incorporate an augmentation term to allow doubly robust estimation of the average treatment effect on the treated. Unlike the matching estimator of the average treatment effect on the treated, the augmented inverse weighted estimator that results from the importance sampling approach has regular asymptotic properties and does not result in any datapoints being excluded from the estimation. Following simulations, we apply the doubly robust, augmented weighted estimator to a U.S. national survey of health to examine the impact of smoking on sleep, and we use techniques developed for other doubly robust estimators to demonstrate model validity.