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Doubly robust (DR) estimation is a crucial technique in causal inference and missing data problems.We propose a novel Propensity score Augmented Doubly robust (PAD) estimator to enhance the commonly used DR estimator for average treatment effect on the treated (ATT), or equivalently, the mean of the outcome under covariate shift.Our proposed estimator attains a lower asymptotic variance than the conventional DR estimator when the propensity score (PS) model is misspecified and the outcome regression (OR) model is correct while maintaining the double robustness property that it is valid when either the PS or OR model is correct.These are realized by introducing some properly calibrated adjustment covariates to linearly augment the PS model and solving a restricted weighted least square (RWLS) problem to minimize the variance of the augmented estimator.Both the asymptotic The two authors have equal contribution.1
Lyu et al. (Fri,) studied this question.