Triply robust estimation under missing at random

Missing data is frequently encountered in many areas of statistics. Imputation and propensity score weighting are two popular methods for handling missing data. These methods employ some model assumptions, either the outcome regression or the response propensity model. However, 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. Thus, we achieve triply robust estimation by adding the outlier robustness condition to the double robustness condition. 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.