Inferences For Semiparametric Varying Coefficient Partially Linear Models

The semiparametric varying coefficient partially linear model, which containsboth the parametric components and the nonparametric components, is more ffexible and explicable than pure parametric and nonparametric regression models.This model is an extension of the linear model, partially linear model and varyingcoefficient model, and has been widely studied in many fields, including economics,biomedical sciences and epidemiology. Moreover, in practice, we often encountersome complicated data such as longitudinal data, missing data and measurementerror data. Hence, it has theoretical and practical significance to study the semiparametric varying coefficient partially linear model with complicated data, whichis becoming one of the hot issues to be discussed in the modern statistical analysis.In this dissertation, we mainly consider inferences for the semiparametricvarying coefficient partially linear model with complicated data such as longitudinal data, missing data and measurement error data. Firstly, we investigate theestimation for the semiparametric varying coefficient partially linear model withcomplicated data based on the empirical likelihood method, which extends theapplication areas of the empirical likelihood method. Secondly, we investigatethe test for the semiparametric varying coefficient partially linear model based onthe profile empirical likelihood method, which modifies the existing testing methods. Lastly, we investigate the variable selection for the semiparametric varyingcoefficient partially linear model with complicated data based on penalized estimation methods, which extends the existing variable selection procedures. More specifically, the research contents of this dissertation are summarized as follows:For the semiparametric varying coefficient partially linear model with independent data, we mainly focus on the test and the variable selection for the parametric components and the nonparametric components. Firstly, based on theprofile empirical likelihood method, the test statistics for the parametric components and the nonparametric components are constructed. Our testing method isdifferent from the existing empirical likelihood test method, because the Wilks'phenomena of our test statistics are still hold, then the rejection regions are constructed. Simulation results show that the test is very sensitive to the alternativehypothesis. Secondly, we present a variable selection procedure by combining basis function approximations with penalized least-squares. Our variable selectionprocedure can select the significant variables in the parametric components andthe nonparametric components simultaneously, which is an essential improvementover the existing variable selection methods. Furthermore, considering that theproperties of the regularized estimators depend on the penalty function, we improve the choice the tuning parameters in the penalty function. Simulation resultsshow that the proposed variable selection procedure performs well.For the semiparametric varying coefficient partially linear model with longitudinal data, we propose a groupwise empirical likelihood procedure that canhandle the inter-series dependence of the longitudinal data. Furthermore, by correcting the attenuation, we propose a corrected empirical likelihood procedure forthe semiparametric varying coefficient partially linear EV model with longitudinaldata. By using residual-adjustment, an empirical likelihood ratio function for thenonparametric components is also constructed. We proved that, in each case, the empirical likelihood ratio functions for parametric components and the nonpara-metric components are all asymptotically chi-squared without undersmoothing.Based on this result we can construct the confidence regions of the parametriccomponents and the pointwise confidence intervals of the nonparametric compo-nents, and the existing data-driven algorithm is valid for selecting an optimalbandwidth in our estimation procedure. A simulation study and an analysis of areal data are undertaken to assess the finite sample performance of the proposedmethods.For the semiparametric varying coefficient partially linear model with missingresponses at random, we consider the empirical likelihood inferences for the parametric components. Based on the inverse marginal probability weighted approach,and by constructing the imputation based auxiliary random vector ingeniously, anempirical likelihood ratio function with imputed values are constructed. We showthat the empirical likelihood ratio function is asymptotically chi-squared that is anessential improvement over existing imputation based estimation methods. Furthermore, we present a variable selection procedure based on penalized estimatingequations. With appropriate selection of the tuning parameters, we show thatthe proposed variable selection procedure can identify the true model consistently,and the regularized estimators achieve the optimal convergence rate, Simulationresults show that the proposed empirical likelihood method and the variable selection procedure with imputed data perform well.For the semiparametric varying coefficient partially linear EV model, wemainly focus on the variable selection when the covariates in the parametric components and the nonparametric components are all measured with errors. A bias corrected variable selection procedure is proposed by combining bias correctionwith penalized least-squares. This variable selection procedure allows that thecovariates in the parametric components and the nonparametric components areall measured with errors, which is very different from the existing variable selection procedures. Furthermore, we also propose a new bias correction for thenonparametric components in our variable selection procedure. With appropriateselection of the tuning parameters, we show that this variable selection procedurecan identify the true model consistently, and the regularized estimators have oracleproperty. Simulation results show that the proposed method is workable.For the covariate adjusted linear regression model, a corrected empiricallikelihood inference is also investigated. Based on the empirical likelihoodmethod, a corrected empirical likelihood ratio function is proposed and theWilks'phenomenon is derived. Then, the confidence intervals for the regressioncoefficients are constructed. Our method does not require consistent estimatorsof the distorting functions and the asymptotic variance, which is a substantial improvement over existing methods. A simulation study and a real data applicationare undertaken to assess the finite sample performance of the proposed method.