Empirical likelihood based weighted GMM estimation with missing response at random

Consider the estimating problem for the parameters defined by general estimating equations in the presence of missing responses. Under the assumption that the propensity score is known up to an unknown parameter vector, we define two estimators of the interest parameter vector using weighted generalized method of moment (GMM) with the weights derived by empirical likelihood with a dimension reduction constraint. The dimension reduction constraint is used to supply the auxiliary information of covariates. This avoids the calculation unmanageable when the dimensionality of covariate vector is high. Generally, the asymptotic variances of the two estimators cannot achieve the semiparametric efficient bound except for special cases. The second estimator is an improved one of the first estimator since the second one has smaller asymptotic variance. Asymptotic theories for the proposed estimators are established. A simulation study was conducted to compare the proposed methods with the existing ones and the simulation results suggest that the proposed approach performs very competitively especially in the case where the dimensionality of covariate vector is high.