Robust estimation of distribution functions and quantiles with non-ignorable missing data

This paper considers several robust estimators for distribution functions and quantiles of a response variable when some responses may not be observed under the non-ignorable missing data mechanism. Based on a particular semiparametric regression model for non-ignorable missing response, we propose a nonparametric/semiparametric estimation method and an augmented inverse probability weighted imputation method to estimate the distribution function and quantiles of a response variable. Under some regularity conditions, we derive asymptotic properties of the proposed distribution function and quantile estimators. Two empirical log-likelihood functions are also defined to construct confidence intervals for distribution function of a response variable. Simulation studies show that our proposed methods are robust. In particular, the semiparametric estimator is more efficient than the nonparametric estimator, and the inverse probability weighted imputation estimator is bias-corrected.