Augmented inverse probability weighted estimation for conditional treatment effect

The Augmented Inverse Probability Weighted (AIPW) estimation has been extensively studied in various scenarios across diverse research areas. This investigation explores the asymptotic properties of the AIPW estimator for the conditional average treatment effect. Under various combinations of the parametric, semiparametric, and nonparametric structure in the nuisance propensity score and outcome regression models, we discuss the asymptotic bias and compare the asymptotic variances among the corresponding estimators. The study covers the asymptotic properties with no model misspecified; with either propensity score or outcome regressions locally/globally misspecified; and with all models locally/globally misspecified. To provide a deeper insight into the nature of these estimators and out of curiosity, we reveal the phenomenon that the asymptotic variances, with model-misspecification, could sometimes be even smaller than those with all models correctly specified. We also conduct a numerical study to validate the theoretical results.