Local polynomial fitting in nonparametric regression

Local polynomial fitting is an attractive method in nonparametric estimation both from theoretical and practical points of view. In this thesis, based on local polynomial fitting, for univariate nonparametric regression, we will provide the asymptotic distribution for the maximum of the normalized deviation of the estimate of the regression function from the true regression function. Using this result we will construct the confidence interval for true regression function.;Varying-coefficient models are useful extensions of the classical linear models. The appeal of these models is that the coefficient functions can be estimated via a simple local polynomial fitting to yield a one-step estimator. In this thesis, we show that the one-step estimators of the coefficient functions are asymptotic normal. The residual variance of the one-step estimator is also estimated and its asymptotic normality is established. Moreover, we consider the important issue of how to select the bandwidth. In the case of estimating coefficient functions, a variable bandwidth is desirable, and which should be indicated by the data themselves. We propose a data-driven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. Some examples are reported to illustrate our bandwidth selection procedure. Moreover, focus on a variable bandwidth selection procedure, we provide the conditional bias and the conditional variance of the estimator, the convergence rate of the bandwidth, and the asymptotic distribution of its error relative to the theoretical optimal variable bandwidth.;Based on one-step estimate method in estimating the coefficient functions in varying-coefficient model, we provide the asymptotic distribution for the maximum of the normalized deviation of the estimate of the coefficient function from the true coefficient function in this thesis. Using this result we will construct the confidence interval for the true coefficient function. Some simulation results are reported to illustrate our method.;Moreover, we will show that one-step method can not be optimal when different coefficient functions admit different degrees of smoothness in varying-coefficient models. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared errors for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental dataset.;Covariance structure analysis is an important technique in data analysis. Statistical theory and compute program have been developed for this model when data are univariate or multivariate. In practice, sometimes the observations are not just multivariate, but they are sampled from continuous functions. In this thesis, we try to give some method to deal with this kind of data by local polynomial modeling technique. We provide analysis some of covariance structure analysis such as principal component analysis by local polynomial modeling when data are curves. Some simulations are presented to support our method.