Abstract: To estimate the treatment effect in an observational study, we use a semiparametric locally efficient dimension-reduction approach to assess the treatment assignment mechanisms and average responses in both the treated and the non-treated groups. We then integrate our results using imputation, inverse probability weighting, and doubly robust augmentation estimators. Doubly robust estimators are locally efficient, and imputation estimators are super-efficient when the response models are correct. To take advantage of both procedures, we introduce a shrinkage estimator that combines the two. The proposed estimators retains the double robustness property, while improving on the variance when the response model is correct. We demonstrate the performance of these estimators using simulated experiments and a real data set on the effect of maternal smoking on baby birth weight.

Key words and phrases: Average treatment effect, double robust estimator, efficiency, inverse probability weighting, shrinkage estimator.