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.

翻译：缺失数据是统计学许多领域中常见的问题。插补法和倾向性评分加权法是处理缺失数据的两种流行方法。这些方法依赖于某些模型假设，要么是结果回归模型，要么是响应倾向性模型。然而，在存在缺失数据的情况下，正确指定统计模型可能具有挑战性。双重稳健估计具有吸引力，因为当结果回归模型或倾向性评分模型中的任意一个被正确指定时，估计量的一致性得以保证。在本文中，我们首先利用信息投影，在间接模型校准约束下开发一种高效且双重稳健的估计量。通过施加内部偏差校准条件来估计回归参数，所得的倾向性评分估计量可以等价地表示为双重稳健的回归插补估计量。此外，我们推广了信息投影，以允许针对异常值进行稳健估计。因此，通过将异常值稳健性条件添加到双重稳健性条件中，我们实现了三重稳健估计。文中给出了一些渐近性质。模拟研究证实，所提出的方法不仅在违反各种模型假设时，而且在存在异常值的情况下，都能实现稳健推断。