Statistical Inference And Applications For Several Kinds Of Varying-Coefficient Models

This paper primarily studies the problems of statistical inference and applications in sev-eral kinds of varying-coefficient models, including the estimation of varying-coefficient models with discontinuous coefficient functions, the detection of jumps and estimation of coefficients in time-varying coefficient models, the orthogonality-projection-based estimation in semi-varying coefficient models with heteroscedastic errors, testing for heteroscedasticity in semi-varying co-efficient models and the estimation in semi-varying coefficient models with longitudinal data. The main contents are arranged as follows:Chapter 1 is devoted to introducing the research background, significance, status and the existing problems for several kinds of varying-coefficient models. In addition, an outline of this paper about our major work is given, encompassing innovative keys.Chapter 2 focuses on the estimation of varying-coefficient models with discontinuous coeffi-cient functions. Based on local linear smoothing and jump-preserving regression techniques, an adaptive jump-preserving estimation method is proposed to estimate the coefficient functions with jumps, which can automatically accommodate possible jumps of the coefficient functions without knowing the number and location of jump points and performing any hypothesis test-s. Under some mild conditions, the asymptotical properties of the resulting estimators can be established. Furthermore, several numerical studies are conducted to evaluate the finite sample performance of the proposed methodologies. Finally, an application with real data illustrates the usefulness of the proposed techniques.Chapter 3 is devoted to studying the detection of jump points and estimation of coefficient coefficients in time-varying coefficient models. In certain applications, the underlying coeffi-cient curves may have singularities, including jumps at some unknown positions, representing structural changes of the related process. Detection of such singularities is important for un-derstanding the structural changes. An alternative jump detection method is proposed based on the local polynomial technique and zero-crossing properties of second-order derivative of a coefficient curve. Using the detected jumps, a coefficient curve estimation procedure is also proposed, which can preserve possible jumps well when the noise level is small. Further, the practical problem of implementation for procedure parameters is discussed. Under some mild conditions, the asymptotic properties of the proposed estimators are established not only in the continuous regions of coefficient functions but also in the neighborhoods of the jumps. Finally, results of two Monte Carlo experiments are presented to examine the finite sample performances of the proposed procedures and two empirical examples are discussed.Chapter 4 is concerned with the orthogonal estimation in semi-varying coefficient models with heteroscedastic errors. Based on the orthogonal projection of a matrix, local linear estima-tion and weighted least squares estimation, an iterated two-stage orthogonality-projection-based estimation is proposed. This method can easily be used to estimate parameters, nonparametric curves, and the variance function. The resulting estimators for parameters and nonparametric curves do not affect each other. Under some mild conditions, the consistency and asymptotic normality for these estimators are studied explicitly. Moreover, some simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by a real data set.Chapter 5 investigates the problem of detecting heteroscedasticity in semi-varying coeffi-cient models. It is important to detect the variance heterogeneity in regression model because efficient inference requires that heteroscedasticity is taken into consideration if it really exists. In this chapter, we present two classes of tests of heteroscedasticity for semi-varying coefficient models. The first test statistic is constructed based on the residuals, in which the error term is from a normal distribution. The second one is motivated by the idea that testing heteroscedas-ticity is equivalent to testing pseudo-residuals for a constant mean. Asymptotic normality is established with different rates corresponding to the null hypothesis of homoscedasticity and the alternative. Some Monte Carlo simulations are conducted to investigate the finite sample performance of the proposed tests. The test methodologies are illustrated with a real data set example.Chapter 6 is concerned with the estimation in semi-varying coefficient models for longitu-dinal data. Semiparametric smoothing methods are usually used to model longitudinal data, and the interest is to improve efficiency for regression coefficients. By the QR decomposition, local linear technique, quasi-score estimation, and quasi-maximum likelihood estimation, we propose a two-stage orthogonality-based method to estimate parameter vector, coefficient func-tion vector, and covariance function. The developed procedures can be implemented separately and the resulting estimators do not affect each other. Under some mild conditions, asymptotic properties of the resulting estimators are established explicitly. In particular, the asymptotic behavior of the estimator of coefficient function vector at the boundaries is examined. Further, the finite sample performance of the proposed procedures is assessed by Monte Carlo simulation experiments. Finally, the proposed methodology is illustrated with an analysis of an acquired immune deficiency syndrome (AIDS) data set.