Estimation and inference based on Neumann series approximation to locally efficient score in missing data problems.

Theory on semiparametric efficient estimation in missing data problems has been systematically developed by Robins and his coauthors. Except in relatively simple problems, semiparametric efficient scores cannot be expressed in closed forms. Instead, the efficient scores are often expressed as solutions to integral equations. Neumann series was proposed in the form of successive approximation to the efficient scores in those situations. Statistical properties of the estimator based on the Neumann series approximation are difficult to obtain and as a result, have not been clearly studied. In this paper, we reformulate the successive approximation in a simple iterative form and study the statistical properties of the estimator based on the reformulation. We show that a doubly-robust locally-efficient estimator can be obtained following the algorithm in robustifying the likelihood score. The results can be applied to, among others, the parametric regression, the marginal regression, and the Cox regression when data are subject to missing values and the missing data are missing at random. A simulation study is conducted to evaluate the performance of the approach and a real data example is analyzed to demonstrate the use of the approach.