Quantiie Regression And Variable Selection Of Semiparametric Regression Models With Index

The semi-parametric regression models with index parameters are an important kind regression model in semi-parametric statistical models with multidimensional covariates, which mainly includes single-index model, partial linear single-index model, single-index varying coefficient model and varying coefficient single-index model. One of the key prop-erties of these models is that they can avoid the so-called " curse of dimensionality" by reducing multidimensional covariates to a univariate index predictor while still capture the underling relationship between the response variable and the multidimensional covari-ates. Because of the above reasons, statistical inference on this kind of models is still an important problem and a hot topic in current statistical research.Existing estimation methods of these models mostly focus on the mean regression, based on the least square method or likelihood related methods. However, these methods are sensitive to outliers and their efficiency may be significantly reduced when the model error’s distribution deviates from normal. In contrast the mean regressing only describe the average of the response, quantile regression proposed by Koenker and Basset [29] can provide more valuable information of the distribution of the response and avoid the influence of the outliers. Since the estimation efficiency of quantile regression is fluctuating with the particular value of the quantile, Zou and Yuan [72] proposed composite quantile regression method, which can combine the information provided by the multiple quantile regression and overcome the estimation efficiency depressing for single quantile regression. The composite quantile regression has been proved to be able to overcome the influence of the non-normal model error effectively and improve the estimation efficiency enormously.There have been some initial studies on the quantile regression and composite quantile regression of the parametric model with index parameter. Wu et al.[55] consider the quantile regression of single-index model and Jiang et al.[23] extended the method by Wu et al.[55] to composite quantile regression of SIM. However, there has been no research on the quantile regression or composite quantile regression on the other two models until now. In this thesis, we mainly consider the quantile regression(QR), composite quantile regression(CQR) and variable selection of the single-index model, partial linear single-index model and single-index varying coefficient model and fill up the blank of the research of the quantile regression and variable selection for the semi-parametric models with index. Firstly, consider the single-index model (SIM),which has form asY=g(XTθ)+ε, where Y∈R is the response variable, X=(X1,…，Xp)T∈Rp is the covariates, ε is the model error, g(·) is unknown link function,θ=(θ1,…,θp)T∈Rp is the unknown index parameter. For model identifiability, we set||θ||=1and θ1>0.We propose a new estimation method to implement the QR and CQR of the SIM, prove the convergence of the algorithm. The estimators obtained achieve the best con-vergence rate. The asymptotic distributions of the proposed estimators are established. Furthermore, by combining the proposed estimation method with adaptive LASSO (Zou [71]), we propose a variable selection methods to select the important predictor in QR and CQR of SIM respectively. The oracle properties of the variable selection methods are also derived. Simulation and real data analysis are conducted to illustrate the good performance of our methods.Secondly, we consider the quantile regression and composite quantile regression of partial linear single-index model, which has standard form asY＝g(XTθ)＋ZTβ+(?) Z=(Z1,…，Zd)T∈Zd)T∈Rd are linear covariates,β is unknown linear parameter, the other conditions is the same as that in above single-index model. We proposed an minimizing average check loss estimation(MACLE) method to get the estimation of the parameter and the nonparametric function g(·), which can reach the best convergence rate and avoid the common "under smoothing" problem in semi-parametric regression. The asymptotic prop-erties of the proposed estimators are established. Furthermore, we combine the MACLE method and the adaptive LASSO method to conduct variable selection in QR estima-tion. The oracle properties of the adaptive LASSO penalized MACLE variable selection method are also established. Since the estimation efficiency of single quantile regression by MACLE fluctuating with the quantile value and the estimation efficiency can be im-proved by combing the information provided by multiple quantile regression, we consider the composite quantile regression of PLSIM. we extend MACLE method to composite quantile regression of PLSIM and propose composite-MACLE(C-MACLE) method. The estimator obtain by C-MACLE are shown to be able to achieve the best convergence rate and asymptotically normal distributed.. The asymptotic relative efficiency (ARE) of C-MACLE to least-square method is also investigated. Further more, we extend the above variable selection method to C-MACLE and establish the responding oracle prop-erty. Simulation and real data analysis are conducted to evaluate the properties of the methods proposed.At last, we consider the quantile regression and composite quantile regression of the single-index coefficient model(SICM), which has form asY=g(XTθ)TZ+∈Z=(Z0,…,Zd-1)T∈Rd are covariates, g(·)=(go(·),g1(·),…gq-1(·))T are unknown coefficient vector function, the other conditions is the same as that in above single-index model. Without loss generality, we set Z0=1. Studies of quantile regression of SICM has not been carried out until now. We apply the MACLE and C-MACLE methods to SICM, and establish the asymptotic properties of the estimators, which can achieve the best convergence rate. In addition, we compare the asymptotic relative efficiency between C-MACLE and the profile likelihood estimation method proposed by Lu et al.[36]. More-over, we propose to combine the MACLE and C-MACLE method with adaptive LASSO penalized estimation method to conduct variable selection and derive the responding or-acle property. Random simulation with various errors and real data analysis confirm the good properties of our methods.