Empirical Likelihood For Quantile Regression With Censored Covariates

Censored data is the incomplete data that is not completely observable in a given setting.Its emergence has caused a series of problems,such as the data processing and the analysis becomes complex.How to choose the appropriate method to analyze the incomplete data sets is always the problem of data processing.The simplest approach is to remove the censored data,but this method can lead to large deviations.Rubin(1977)system induces the idea and theoretical framework of multiple imputation methods,and realizes the multiple imputation of incomplete data by simulation,avoiding the information loss caused by directly deleting the incomplete data.In the current literature,many scholars pay attention to the responses censored at random,but few studies have been given on the censored covariates.Based on the relevant literatures,we considered the empirical likelihood for quantile regression models with censored covariates.Firstly,the conditional density of censored covariance is estimated based on the method of the quartile regression,and we use the multiple imputation approach to fill the censored covariates.Then,constructed the empirical likelihood ratio statistic of regression parameters,and show that the proposed empirical log-likelihood ratios are both asymptotically Chi-squared in theory.And construct the confidence regions for the parametric.Finally,we compared the results of the multiple imputation method(IEL)and the method of deleting the censored data directly(CEL).The main structure of this paper is organized as follows:the statement in the first two chapters focuses on the present situation of domestic and foreign research and the preliminary knowledge,and gives a detailed theoretical introduction to the censored quantile regression,multiple interpolation and empirical likelihood.The third chapter introduces the multiple imputation method based on quantile regression,and discusses how to construct the corresponding empirical likelihood confidence region.Based on the above method discussed,a simulation experiment was carried out.We consider homoscedastic model and heteroscedastic model.First we use the multiple imputation method to fill the censored data,then we compared the length of empirical likelihood average confidence interval obtained by the multiple imputation method with the average interval length obtained by the method of directly deleting the censored data,and simulation results indicate that the proposed method is workable.Finally,the proposed method is applied to a C-reactive protein dataset in the 2007-2008 National Health and Nutrition Examination Survey.Simulation and empirical results show that the multiple imputation method(IEL)in this paper has smaller standard error and shorter average confidence interval than directly deleting the censored data(CEL).Therefore,compared with the traditional method,the data processing method proposed in this paper is effective and feasible.