  Missing data is frequently encountered in many areas of statistics.
Propensity score weighting is a popular method for handling missing data. The
propensity score method employs a response propensity model, but correct
specification of the statistical model can be challenging in the presence of
missing data. Doubly robust estimation is attractive, as the consistency of the
estimator is guaranteed when either the outcome regression model or the
propensity score model is correctly specified. In this paper, we first employ
information projection to develop an efficient and doubly robust estimator
under indirect model calibration constraints. The resulting propensity score
estimator can be equivalently expressed as a doubly robust regression
imputation estimator by imposing the internal bias calibration condition in
estimating the regression parameters. In addition, we generalize the
information projection to allow for outlier-robust estimation. Some asymptotic
properties are presented. The simulation study confirms that the proposed
method allows robust inference against not only the violation of various model
assumptions, but also outliers. A real-life application is presented using data
from the Conservation Effects Assessment Project.
