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.

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.

In this paper, the varying-coefficient single-index model (VCSIM) is proposed. It can be seen as a generalization of the semivaryingcoefficient model by changing its constant coefficient part to a nonparametric component, or a generalization of the partially linear single-index model by replacing the constant coefficients of its linear part with varying coefficients. Based on the local linear method, average method and backfitting technique, the estimates of the unknown parameters and the unknown functions of the VCSIM are obtained and their asymptotic distributions are derived. Both simulated and real data examples are given to illustrate the model and the proposed estimation methodology.

In this article, a procedure for estimating the coefficient functions on the functional-coeffcient regression models with different smoothing variables in different coefficient, functions is defined. Firs step, by the local linear technique and the averaged method, the initial estimates of the coefficient functions are given. Second step, based on the initial estimates, the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure. The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions. Two simulated examples show that the procedure is effective.