Abstract: Handling data with the missing not at random (MNAR) mechanism is still a challenging problem in statistics. In this article, we propose a nonparametric imputation method based on the propensity score in a general class of semiparametric models for nonignorable missing data. Compared with the existing imputation methods, the proposed imputation method is more flexible as it does not require any model specification for the propensity score but rather a general parametric model involving an unknown parameter which can be estimated consistently. To obtain a consistent estimator of the parametric propensity score, two approaches are proposed. One is based on a validation sample. The other is a semi-empirical likelihood (SEL) method. By incorporating auxiliary information from some calibration conditions under the MNAR assumption, we gain significant efficiency with the SEL-based estimator. We investigate the asymptotic properties of the proposed estimators based on either known or estimated propensity scores. Our empirical studies show that that the resultant estimator is robust against the misspecified response model. Simulation studies and data analysis are provided to evaluate the finite sample performance of the proposed method.

Key words and phrases: Generalized method of moments, imputation, not missing at random, propensity score, semi-empirical likelihood, semi-parametric estimation.