Robust Frequentist Model Averaging And Estimation Of Semiparametric Regression Models

This dissertation is concerned with the robust frequentist model averagingand estimation of semiparametric isotonic regression models.Firstly, model selection and model averaging for quantile regression, rank re-gression and M-estimation with local misspecification framework are considered,respectively. Under some regularity conditions, asymptotic normality of the esti-mators of regression coefcients for each submodel are presented. Based on theasymptotic normality, the asymptotic properties of estimators of focus parametersare discussed. Using the asymptotic mean squares error of estimators of focusparameters, FICs are developed for the three regressions respectively. In addition,we obtain the asymptotic distributions of general frequentist model averaging es-timators, and construct a proper confidence interval. Simulations are carried outto illustrate the performance of the proposed procedures. These works are dividedinto Chapter2to Chapter4.Secondly, in Chapter5for censored quantile regression models, we proposethe FIC and frequentist model averaging method. Asymptotic normality of theestimators of regression coefcients for each submodel is presented by inverse prob-ability weight method. Meanwhile, we discuss the asymptotic properties of thesubmodel estimator and general model averaging estimator of the focus parameter.A simulation study is carried to examine finite sample properties of the proposedmethods, and a real data analysis is given for illustration.Thirdly, variable selection for partially linear varying coefcient quantile re- gression models is considered. The functional coefcients are estimated by basisfunction approximations, and thus the partially linear varying coefcient quantileregression models are transformed into “linear regression models”. Based on penal-ized quantile loss function, variable selection is proposed. We further demonstratethat, with proper choices of the penalty functions and the regularization parame-ter, the resulting estimates perform asymptotically as well as an oracle estimator,and the functional coefcients have the optimal convergence rate. Simulation re-sults and real data analysis show that the proposed procedure performs well.Fourthly, in Chapter7, variable selection for semiparametric isotonic regres-sion models is considered. The method based on the kernel estimation and penal-ized least square is employed to gain the spare estimator of the parameter basedon which the estimator of the functional component are obtained further. Withappropriate selection of the tuning parameters, we show that the regularized es-timators have oracle property. The estimator of the nonparametric componentis monotone and, at a fixed point, is shown to be cubic root-n consistent andconverges in distribution to the left slop at zero of the greatest convex minorantof the sum of a two-sided standard Brownian motion and the square of the timeparameter. Simulation results show that the proposed procedure performs well.Finally, M-estimation for semparametric isotonic regression models is con-sidered in Chapter8. The functional component is estimated by basis functionapproximations. Under general loss function, by virtue of the empirical processtheory, the estimator of the parameter is shown to be root-n consistent and asymp-totic normal, and the estimator of the nonparametric component not only has theoptimal rate of convergence, but also monotone. A simulation study and an anal- ysis of a real data are undertaken to assess the finite sample performance of theproposed methods.