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.

To perform hydrological forecasting, time series methods are often employed. In univariate time series, the autoregressive integrated moving average (ARIMA) model, the seasonal autoregressive moving average (SARMA) model, the deseasonalized model and the periodic autoregressive (PAR) model are often used. These models are based on the assumption that the influence of lagged riverflows on the riverflow is linear. In reality the assumption is often questionable. In this paper, the functional-coefficient autoregression (FCAR) model, which is a nonlinear model, is introduced to forecast riverflows. To explore the influence of the inflow on the outflow in a river system and to exploit the internal interaction of the outflows, bivariate time series models are needed. The transfer function (TF) model and the semi-parametric regression (SPR) model are often employed. In this paper, a new model, the non-parametric and functional-coefficient autoregression (NFCAR) model, is proposed. It consists of two parts: the first part, the non-parametric part explains the influences of the inflows on the outflow in a river system; the second part, the functional-coefficient linear part reveals the interactions among the outflows in a river system. By comparing the calibration and forecasting of the models, it is found that the NFCAR model performs very well.

Varying-coefficient models are popular multivariate nonparametric fitting techniques. When all coefficient functions in a varyingcoefficient model share the same smoothing variable, inference tools available include the F-test, the sieve empirical likelihood ratio test and the generalized likelihood ratio (GLR) test. However, when the coefficient functions have different smoothing variables, these tools cannot be used directly to make inferences on the model because of the differences in the process of estimating the functions. In this paper, the GLR test is extended to models of the latter case by the efficient estimators of these coefficient functions. Under the null hypothesis the new proposed GLR test follows the 2-distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Further, we have derived its asymptotic power which is shown to achieve the optimal rate of convergence for nonparametric hypothesis testing.A simulation study is conducted to evaluate the test procedure empirically.

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.

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.