Statistical Methods And Theoretical Research Of A Class Of Varying Coefficient Models

Varying coefficient model is a class of very important nonparametric regres-sion model,statistical inference of this model is one of the hot topics of the modern statistical analysis. Compared to the classic linear model, varying coefficient model has stronger adaptability and modeling capability for the regression coefficients are nonparametric functions. At the same time, varying coefficient model still has the linear structure, thus it can easily be understand and explain in practical ap-plication. Moreover, it is flexible enough to cover many important models, such as partially linear model, partially linear varying coefficient model, additive model, linear model, and so on. Hence, varying coefficient model has a wide range of appli-cation in biomedical, econometrics, environmental science, and is a powerful tool for dealing with complicated data. In this dissertation, we focus on inference for a class of varying coefficient models, including varying coefficient model, partially linear varying coefficient model, partially varying coefficient single index model, single index varying coefficient model, partially varying coefficient functional lin-ear model, etc. More specifically, the research contents of this dissertation are summarized as follows:For the partially liner varying coefficient model with measurement error in the nonparametric part, when some prior information about the parametric part is available, two types of local bias-corrected restricted profile least squares estima-tors of the parametric component and nonparametric component are conducted, and their asymptotic properties are also studied under some regularity conditions. Moreover,we compare the efficiency of the two kinds of parameter estimators un-der the criterion of Lowner ordering. Finally, we construct a local bias-corrected profile Lagrange multiplier test statistic for the unknown parametric component, and show that its limiting distribution is a standard chi-squared distribution under the null hypothesis. Some simulation studies and an analysis of a real data are undertaken to assess the finite sample performance of the proposed methods.For the partially varying coefficient single-index model, we propose a regu-larized variable selection procedure by combining basis function approximations with SCAD penalty. The proposed procedure simultaneously selects significant variables in the single-index parametric components and the nonparametric coeffi-cient function components. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. Moreover, the proposed method can naturally be applied to deal with pure single-index model and varying-coefficient model. Simu-lation study and real data example show that the proposed procedure is workable.For the single index varying coefficient model, firstly, we focus on the variable selection for the parametric components and the nonparametric components. We present a variable selection procedure by combining basis function approximations with penalized least squares. Our variable selection procedure can select the sig-nificant covariates with functional coefficients and local significant variables with parametric coefficients simultaneously. With appropriate selection of the tuning parameters, we establish the consistency of the variable selection procedure and the oracle property of the regularized estimators. Secondly, we consider the prob-lem of model detection and estimation for single index varying coefficient model, and propose a new combined penalization procedure by penalizing the coefficient functions and their derivatives simultaneously. Based on the proposed combined penalization procedure,we can identify the true model structure consistently, and obtain a new semiparametric model-partially linear single-index varying coeffi-cient model(PLSIVCM). Under the appropriate conditions, we demonstrate that the proposed penalized estimators of parametric and nonparametric components of PLSIVCM are consistent, but their asymptotic distributions are not available. Hence, we extend the minimum average variance estimation (MAVE) method to PLSIVCM, and establish the asymptotic normality for the refined estimators of index parameters, constant coefficients and varying coefficient functions, re-spectively. Monte Carlo simulation study and real data example show that the proposed methods are workable.For the partially varying coefficient functional linear model, we mainly con-sider the problem of estimation and test for the unknown coefficient functions. Firstly, by means of functional principal components analysis and local linear smoothing techniques, we obtain the estimators of coefficient functions of both function-valued variable and real-valued variables. Then the rates of convergence of the proposed estimators are established under some regularity conditions. Sec-ondly, in order to test the existence of interaction effects among the real-valued explanatory variables, we develop a hypothesis test for the model and employ the bootstrap procedure to evaluate the null distribution of test statistic and the p-value of the test. Finally, we apply the proposed model and methods to predict the fat content of a meat sample from the Tecator data, and show that the proposed methods perform well.At last, we construct tests for heteroskedasticity in nonparametric time-varying coefficient panel data models with fixed effects. Firstly, based on the local linear smoothing technique, we obtain the estimators of the unknown co-efficient functions and model residuals. Secondly, when T tends to infinite or T is fixed, we present different artificial regressions and test statistics to test the existence and source of heteroskedasticity in the model. The proposed tests are distribution free and can easily be implemented. Some simulated examples show that our proposed methods perform well.