Efficient and doubly robust estimation in covariate-missing data problems

Missing covariate data often arise in the health and social sciences as well as in survey sampling. By employing a working propensity score and a working regression function, Robins, Rotnitzky, and Zhao (1994) introduced the augmented inverse probability weighting (AIPW) method for estimation of regression parameters, which extends the inverse probability weighting (IPW) method of Horvitz and Thompson (1952); the AIPW estimators are locally efficient and doubly robust. Qin, Zhang, and Leung (2009) proposed empirical likelihood estimators of regression parameters by using the empirical likelihood method to combine a set of estimating equations; their estimators are locally efficient and are more efficient than the AIPW estimators when the working propensity score is correctly specified, but it is unclear whether they are doubly robust. In this paper, we propose a hybrid of the empirical likelihood method and the method of moments, which can generate estimators for regression parameters that enjoy three features: (a) they are asymptotically equivalent to those of Qin, Zhang, and Leung (2009) when the working propensity score model is correctly specified, (b) they are more efficient than the AIPW estimators when the working propensity score is correct, and (c) they are doubly robust. Since the proposed hybrid method imposes a smaller number of constrains, it is also computational less intensive than the empirical likelihood method of Qin, Zhang, and Leung (2009). We present a simulation study to compare the finite-sample performance of various methods with respect to bias, efficiency, and robustness to model misspecification.