DOUBLY ROBUST AND LOCALLY EFFICIENT ESTIMATION WITH MISSING OUTCOMES

We consider parametric regression where the outcome is subject to missingness. To achieve the semiparametric efficiency bound, most existing estimation methods require the correct modeling of certain second moments of the data, which can be very challenging in practice. We propose an estimation procedure based on the conditional empirical likelihood (CEL) method. Our method does not require us to model any second moments. We study the CEL-based inverse probability weighted (CEL-IPW) and augmented inverse probability weighted (CEL-AIPW) estimators in detail. Under some regularity conditions and the missing at random (MAR) mechanism, the CEL-IPW estimator is consistent if the missingness mechanism is correctly modeled, and the CEL-AIPW estimator is consistent if either the missingness mechanism or the conditional mean of the outcome is correctly modeled. When both quantities are correctly modeled, the CEL-AIPW estimator attains the semiparametric efficiency bound without modeling any second moments. The asymptotic distributions are derived. Numerical implementation through nested optimization routines using the Newton-Raphson algorithm is discussed.