This paper considers estimating a parameter β that defines an estimating function U(y,x; β)$ for an outcome variable y and its covariate x when the outcome is missing in some of the observations. We assume that, in addition to the outcome and the covariate, a surrogate outcome is available in every observation.

The efficiency of existing estimators for β depend critically on correctly specifying the conditional expectation of U given the surrogate and the covariate. When the conditional expectation is not correctly specified, which is the most likely the case in practice, the estimation efficiency can be severely compromised. We propose an estimator that is robust against the choice of the conditional expectation via the empirical likelihood.

We demonstrate that the proposed estimator achieves efficiency gain whether the conditional score is correctly specified or not. When the conditional score is correctly specified, the estimator reaches the semi-parametric variance bound within the class of estimating functions generated by U. The practical performance of the estimator is evaluated using simulation and a dataset based on the 1996 U.S. presidential election.