The adaptive varying-coefficient partially linear regression (AVCPLR) model is proposed by combining the nonparametric regression model and varying-coefficient regression model with different smoothing variables. It can be seen as a generalization of the varying-coefficient partially linear regression model, and it is also an example of a generalized structured model as defined by Mammen and Neilsen [Mammen, E., Nielsen, J.P., 2003. Generalised structured models. Biometrika 90, 551 566]. Based on the local linear technique and the marginal integrated method, the initial estimators of these unknown functions are obtained, each of which has big variance. To decrease the variances of these initial estimators, the one-step backfitting technique proposed by Linton [Linton, O.B., 1997. Efficient estimation of additive nonparametric regression models. Biometrika 82, 93 100] is used to obtain the efficient estimators of all unknown functions for the AVCPLR model, and their asymptotic normalities are studied. Two simulated examples are given to illustrate the AVCPLR model and the proposed estimation methodology.

In this paper, we study a new class of semiparametric models, termed as the proportional functional-coefficient linear regression models for time series data. The model can be viewed as a generalization of the functional-coefficient regression models but is has different proportional functions of parameter and different smoothing variables in the same coefficient function in different position. When the parameter is known, the local linear technique is employed to give the initial estimator of the coefficient function in the model, which does not share the optimal rate of convergence. To improved its convergent rate, a one-step backfitting technique is used to obtain the optimal estimator of the coefficient function. The asymptotic properties of the proposed estimators are investigated. When the parameter is unknown, the method of estimating parameter is given. It can be shown that the estimator kf the parameter is √n-consistent. The bandwidths and the smoothing variables are selected by a data-driven method. A simulated example with two cases and two real data examples are used to illustrate the applications of the model.

Empirical-likelihood-based inference for the nonparametric parts in semiparametric varying-coefficient partially linear (SVCPL) models is investigated. An empirical loglikelihood approach to construct the confidence regions/intervals of the nonparametric parts is developed. An estimated empirical likelihood ratio is proved to be asymptotically standard 2-limit. A simulation study indicates that, compared with a normal approximation-based approach and the bootstrap method, the proposed method described herein works better in terms of coverage probabilities and average areas/widths of confidence regions/bands. An application to a real data set is illustrated.

It is well known that the exposure to ambient air pollution might cause serious respiratory illnesses and that the weather conditions may also contribute to the seriousness. However, quantifying the effects of pollution and the weather condition is a difficult task due to the nonlinear nature of these impacts. The problem is further complicated by the possibly cumulative effects of these impacts. In this paper, the nonparametric additive (NPA) models, which have the advantage of ease in interpretation and forecasting, are employed for modeling the effects of pollution and weather. All models are derived by the local linear method. The variables in the final selected NPA model are chosen by cross-validation method together with bootstrap test for the data of Hong Kong. For comparison the final selected linear regression (LR) model by the backward elimination method is also considered. It is found, interestingly, that the variables selected by nonparametric method and the usual backward elimination method for linear models are different. Furthermore, by comparing forecasted values obtained from the NPA and LR models and true values the final selected NPA model is shown to outperform the LR model.

The functional-coefficient regression models with different smoothing variables in different coefficient functions are discussed in this paper. The averaged estimates of coefficient functions are defined by averaging the sample on the initial value obtained by a local linear technique. Their asymptotic normality is studied. The efficiency of the proposed method is shown by a simulated example.