Efficient restricted estimators for conditional mean models with missing data

Consider a conditional mean model with missing data on the response or explanatory variables due to two-phase sampling or nonresponse. Robins et al. (1994) introduced a class of augmented inverse-probability-weighted estimators, depending on a vector of functions of explanatory variables and a vector of functions of coarsened data. Tsiatis (2006) studied two classes of restricted estimators, class 1 with both vectors restricted to finite-dimensional linear subspaces and class 2 with the first vector of functions restricted to a finite-dimensional linear subspace. We introduce a third class of restricted estimators, class 3, with the second vector of functions restricted to a finite-dimensional subspace. We derive a new estimator, which is asymptotically optimal in class 1, by the methods of nonparametric and empirical likelihood. We propose a hybrid strategy to obtain estimators that are asymptotically optimal in class 1 and locally optimal in class 2 or class 3. The advantages of the hybrid, likelihood estimator based on classes 1 and 3 are shown in a simulation study and a real-data example. Copyright 2011, Oxford University Press.