Weighted Quantile Averaging Estimation For Linear Regression Models With Non-ignorable Missing Data

In practice,missing data or incomplete data set are frequently encountered due to some reasons.In recent years,more and more statisticians focus on statistical analysis of missing data.The most direct and commonly used method to deal with missing data is to delete individuals with any missing data and conduct complete case analysis.However,in many cases,missing data contains some important information,and thus the complete case analysis may cause the loss of information and make the model estimation inaccurate.In order to overcome the problem of complete case analysis,imputation approach and likelihood method and inverse probability weighting method have been proposed to handle the missing data,in order to get a more accurate model.In the regression analysis,quantile regression has many good properties compared with least squares regression.Quantile regression describes the overall distribution characteristics of data through considering different quantile levels,which is more robust than mean regression analysis.Quantile regression has been widely used and applied in practice.However,the estimation efficiency of quantile regression depends on fixed quantile level which motivates us to consider a weighted quantile average estimator by combining information over multiple quantiles.Under the non-ignorable missing data mechanism,we consider a general parametric propensity score model,and employ the maximum likelihood method and the semi-parametric empirical likelihood method to estimate the unknown parameters defined in the assumed propensity score model.Based on inverse probability weighting approach,this paper combines quantile regression method and the idea of model averaging to develop weighted quantile average estimation procedure for linear regression model with non-ignorable missing data.To improve estimation efficiency,we further develop an empirical likelihoodbased inverse probability weighting approach,by incorporating additional auxiliary information.We conduct extensive simulation studies to illustrate the proposed method.The simulation results imply that,compared with the traditional inverse probability weighted least squares method,the proposed inverse probability weighted quantile averaging approach can produces estimators with smaller bias and is more robust against heavy-tailed error distributions.An empirical study using video playback data is further conducted for illustration.