Generalized method of moments for nonignorable missing data

In this study, we consider the problem of nonignorable missingness in the framework of generalized method of moments. To model the missing propensity, a semiparametric logistic regression model is adopted and we modify this model with nonresponse instrumental variables to overcome the identifiability issue. Under the identifiability conditions, we mitigate the effects of nonignorable missing data through reformulated estimating equations imputed via a kernel regression method, then the idea of generalized method of moments is applied to estimate the parameters of interest and the tilting parameter in propensity simultaneously. Moreover, the consistency and the asymptotic normality of the proposed estimators are established and we find that the price we pay for estimating an unknown tilting parameter is an increased variance for the estimator of population parameters, that is quite acceptable in contrast with validation sample, especially for practical problems. The proposed method is evaluated through simulation studies and demonstrated on a data example.