Research On Statistical Models Based On Composite Quantile Regression Method

Quantile regression compared with mean regression, needs no assumption on error distribution, and thus performs more robust. However, quantile regression procedure can result in an arbitrarily small relative efficiency compared with the mean regression. Composite quantile regression method was proposed to overcome this shortcoming. Missing data and censored data as common phenomenons in real life, have became important contents of statistical research. In this thesis, based on the composite quantile regression method, we consider the parametric and semiparametric models under the framework that the data is incomplete. Specifically, four questions are studied in this thesis.In Chapter 2, based on composite quantile regression and inverse probability weighting approach, we propose new estimation procedures in the linear model when some covariates are missing at random. Moreover,in order to identify the nonzero elements of unknown parameters, the penalized weighted composite quantile regression estimators are proposed. Under some mild conditions, the asymptotic normality, Oracle and Horvitz-Thompson properties of the proposed estimators are established. Simulation results and a real data analysis are provided to examine the performance of our methods.In Chapter 3, based on inverse censoring probability weighting, composite quantile regression approach and SCAD penality, a new variable selection procedure is considered in censored linear model with a diverging number of parameters. Moreover, an iterative algorithm is proposed to minimize the objective function. Under some mild conditions, the ??? consistency and Oracle property of the proposed estimator are proved. Some simulations and a real data example are provided to examine the performance of our procedure.In Chapter 4, we consider a new robust estimation in censored partially linear additive models in which the nonparametric components are approximated by polynomial spline. For identifying the significant variables in the linear part, a regularization procedure based on adaptive lasso is proposed for estimation and variable selection simultaneously. Under some regular conditions, the asymptotic normality and oracle property of the parametric components are established, and the optimal convergence rates of the nonparametric components are obtained. Simulation studies and a real data analysis are presented to illustrate the performance of the proposed estimators.In Chapter 5, based on local linear regression, composite quantile regression and inverse probability weighting approach, new estimation methods are proposed in single-index model with covariates missing at random. Under some regularity conditions, we establish the asymptotic properties of the estimators. For the parametric part, it implies a smaller limiting variance for the WCQR estimator with estimated selection probability than that with true selection probability, that is, the Horvitz-Thompson property holds. For the nonparametric part, the asymptotic properties of the WCQR estimators indicate that it has no effect on the asymptotic variance, whether weights are estimated or not. Simulation studies are presented to illustrate the behavior of the proposed estimators.