A unified theory on empirical likelihood methods for missing data

Efficient estimation with missing data is an important practical problem with many application areas. Parameter estimation under nonresponse is considered when the parameter is defined as a solution to an estimating equation. Using a response probability model, a complete-response empirical likelihood method can be constructed and the non-parametric maximum likelihood estimator can be obtained by solving the weighted estimating equation where the weights are computed by maximizing the complete-response empirical likelihood subject to the constraints that incorporate the auxiliary information obtained from the full sample. Often the constraints are constructed from the working outcome regression model for the conditional distribution of the estimating function given the observation. The proposed method achieves the semi-parametric lower bound when we correctly specify the conditional expectation of the estimating function, regardless of whether the response probability is known or estimated. When the response probability is estimated nonparametrically, the resulting empirical likelihood method automatically achieves the semi-parametric lower bound without specifying the conditional distribution of the estimating function. Asymptotic theories are derived and simulation studies are also presented.