An efficient and doubly robust empirical likelihood approach for   estimating equations with missing data
  This paper considers an empirical likelihood inference for parameters defined by general estimating equations, when data are missing at random. The efficiency of existing estimators depends critically on correctly specifying the conditional expectation of the estimating function given the observed components of the random observations. When the conditional expectation is not correctly specified, the efficiency of estimation can be severely compromised even if the propensity function (of missingness) is correctly specified. We propose an efficient estimator which enjoys the double-robustness property and can achieve the semiparametric efficiency bound within the class of the estimating functions that are generated by the estimating function of estimating equations, if both the propensity model and the regression model (of the conditional expectation) are specified correctly. Moreover, if the propensity model is specified correctly but the regression model is misspecified, the proposed estimator still achieves a semiparametric efficiency lower bound within a more general class of estimating functions. Simulation results suggest that the proposed estimators are robust against misspecification of the propensity model or regression model and outperform many existing competitors in the sense of having smaller mean-square errors. Moreover, using our approach for statistical inference requires neither resampling nor kernel smoothing. A real data example is used to illustrate the proposed approach. 
Tianqing Liu, Xiaohui Yuan, Zhaohai Li and Aiyi Liu
stat.ME