Robust estimation under a semiparametric propensity model for nonignorable missing data

We study the problem of estimating a functional or a parameter in the context where outcome is subject to nonignorable missingness. We completely avoid modeling the regression relation between outcome and co-variates, while allowing the propensity to be modeled by a semiparametric relation where the dependence on covariates is unknown and unspecified. This unknown function in the semiparametric propensity model is not directly estimable from the observed data and is the fundamental challenge in the estimation of the parameter of interest. By carefully analyzing the semiparametric structure of the model, we discover that the estimation of the parameter in the propensity model as well as the functional estimation can be carried out without estimating this unknown function. This phenomenon is especially interesting and extremely helpful in our context, because it overcomes the very obstacle that this unknown function cannot be directly estimated from the observed data. The property of the proposed estimator is established rigorously through theoretical derivations, and supported by simulations and a data application.