Some Researches On Quantile Regression Model With Nonignorable Missing Response

Missing data exists widely in many important fields,such as industrial and agricultural production,drug research and development,epidemiology and census.A series of problems arise along with missing data,such as reduction of sample information,decrease of model efficiency,increase of statistical analysis complexity,which are against data analysis and statistical inference.According to different missing mechanisms,missing data can be divided into three types: missing completely at random,missing at random and missing not at random.The first two kinds of missing mechanisms have nothing to do with missing data itself,which is usually called ignorable missing,while the last one is related to missing data,and it is called nonignorable missing.In the case of missing data,the identifiability issure of parameters often brings great challenge to estimation.The processing method of missing data strongly depends on the missing mechanisms.Misuse of the processing method of missing data will produce great deviation to the statistical results,and lead to incorrect conclusions.Quantile regression has many advantages over the traditional conditional mean regression.It can not only measure the influence of covariates in the center of the distribution,but also describe the influence at the tail of the distribution.It highlights the heterogeneous influence of covariates on the whole conditional distribution of response variable and the local correlation.In addition,the quantile regression model does not need to make specific assumptions on the error distribution.Therefore,in the presence of heavy-tailed errors with unequal variances or/and outliers,quantile regression model is more robust than least squares regression.One can obtain a more complete picture of effects of the covariates on the response variable by considering different quantiles.In real data analysis,the quantile regression model may include many irrelevant covariates,especially for covariates with high dimension.In this case,it is important to find which covariates are relevant for prediction,so that we can obtain better interpretation of the model,and better efficiency of the estimator.A number of approaches have emerged,such as information criterion based methods,penalization or shrinkage based variable selection methods including Lasso,SCAD,Elastic net and many others.This dissertation mainly discusses the penalized empirical likelihood,which combines the standard empirical likelihood with a penalty function.Chapter 1 gives a brief review of nonignorable missing,quantile regression,empirical likelihood and variable selection.Chapter 2 proposes to use nonresponse instrument to handle the identifiability problem,and apply the generalized method of moments to estimate the propensity.With nonignorable missing,a smoothed weighted empirical likelihood estimator is proposed by using kernel smoothing method.With high dimensional covariates,penalized variable selection method is also presented.It can be seen that the penalized smoothed weighted empirical likelihood method can efficiently select significant variables and estimate parameters simultaneously.With a proper choice of tuning parameters,the resulting estimator is consistent and has the oracle property.In Chapter 3,in terms of diverging dimension,it shows that penalized smoothed weighted empirical likelihood method can still work in the estimation for quantile regression model,provided that dimension p increases to infinite at a proper rate as the sample size n tends to infinity.In Chapter 4,motivated by a real data set,partially linear quantile regression model is discussed.This part presents to use B-spline basis functions to approximate the nonparametric function and construct bias-corrected and smoothed estimating equations based on the inverse probability weighting approach.The variable selection in the linear component based on the penalized empirical likelihood is also proposed.Chapter 5concludes this dissertation and illustrates perspects and vision of future work.There are three main innovations of this thesis:(1)This dissertation discusses to use instrument variable to handle with identifiability issue,and apply the generalized method of moments to estimate the unknown nonignorable nonresponse propensity;(2)This dissertation presents to construct a smoothed weighted empirical likelihood estimator,it also proves the log-likelihood ratio based on the smoothed weighted empirical likelihood with true quantile regression parameter is shown to have an asymptotically weighted sum of chi-squares,which can be used to construct confidence regions or test statistical hypotheses;(3)With high dimensional covariates,this dissertation proposes a penalized empirical likelihood approach by combining the empirical likelihood with the SCAD method together.Especially,penalized empirical likelihood can still work even with covariates of a diverging dimension.