Statistical Inference Of Semiparametric Models With Incomplete Data

Incomplete data are often encountered in many practical problems,such as survival analysis,medicine tracking test,reliability life test and so on.Incomplete data bring great difficulties to the use and analysis of data,it is also one of the main reasons for the uncertainty of the information system.How to effectively use these incomplete data information and make statistical inference is of great practical significance.The statistical properties for the nonparametric regression models under complete data have developed fairly well,while the statistical analysis under incomplete data is still a field that has no long history and needs to be further developed.In this thesis,we study the statistical inference problems of semiparametric regression models with incomplete data.The main research work can be explained from the following four aspects:In the chapter II,we address the problem of hypothesis test on response mean with various inequality constraints in the presence of covariates when response data are miss-ing at random.when response data are missing at random,based on the approach of weighted-corrected imputation for the response variable,the weighted-corrected em-pirical likelihood ratio test statistics of the response mean with inequality constraints are constructed.Under some wild conditions,we investigate limiting distributions and asymptotic powers of the proposed empirical likelihood ratio test statistics with auxil-iary information.The results show that the test statistics with auxiliary information are more efficient than that without auxiliary information.A simulation study is un-dertaken to investigate the finite sample performance of the proposed method.Chapter III investigates the problem of testing nonparametric function in partial linear errors-in-variables models with response missing at random.In order to over-come the bias produced by measurement errors,two bias-corrected test statistics based on the quadratic conditional moment method are proposed.The limiting null distri-butions of the test statistics are established respectively and p values can be easily determined which show that the proposed test statistics have similar theoretical prop-erties.Moreover,our tests can detect the alternatives distinct from the null hypothesis at the optimal nonparametric rate for local smoothing-based methods in this area.Simulation studies are conducted to demonstrate the performance of the proposed test methods and the proposed two tests give similar performances.A real data set from the ACTG 175 study is used for illustrating the proposed test methods.In the chapter IV,we focus on the problem of estimation and variable selection for quantile regression(QR)of partially linear model(PLM)where the response is subject to random left truncation.We propose a three-stage estimation procedure for parametric and nonparametric parts based on the weights which are random quantities and determined by the product-limit estimates of the distribution function of truncated variable.The estimators obtained in the second and third stages are more efficient than the initial estimators in the first stage.Furthermore,we propose a variable selection procedure for the QR of PLM by combining the estimation method with the SCAD penalty to get sparse estimation of the regression parameter.The oracle properties of the variable selection approach are established.Simulation studies are conducted to examine the performance of our estimators and variable selection method.In the chapter V,we develop a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data.In order to handle the truncated data,the random weights are introduced and the weighted quantile re-gression(WQR)estimators for nonparametric functions are proposed.To achieve nice efficiency properties,we further develop a weighted composite quantile regression(WC-QR)estimation method for nonparametric functions in varying-coefficient models.The asymptotic properties both for the proposed WQR and WCQR estimators are estab-lished.In addition,we propose a novel Bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms.The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.