Functional-coefficient partially linear regression model. (English) Zbl 1139.62023

Summary: A functional-coefficient partially linear regression (FCPLR) model is proposed by combining nonparametric and functional-coefficient regression (FCR) model. It includes the FCR model and the nonparametric regression (NPR) model as its special cases. It is also a generalization of the partially linear regression (PLR) model obtained by replacing the parameters in the PLR model with some functions of the covariates. The local linear technique and the integrated method are employed to give initial estimators of all functions in the FCPLR model. These initial estimators are asymptotically normal. The initial estimator of the constant part function shares the same bias as the local linear estimator of this function in the univariate nonparametric model, but the variance of the former is bigger than that of the latter. Similarly, initial estimators of every coefficient function share the same bias as the local linear estimates in the univariate FCR model, but the variance of the former is bigger than that of the latter.
To decrease the variance of the initial estimates, a one-step back-fitting technique is used to obtain the improved estimators of all functions. The improved estimator of the constant part function has the same asymptotic normality property as the local linear nonparametric regression for univariate data. The improved estimators of the coefficient functions have the same asymptotic normality properties as the local linear estimates in FCR model. The bandwidths and the smoothing variables are selected by a data-driven method. Both simulated and real data examples related to nonlinear time series modeling are used to illustrate the applications of the FCPLR model.


62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics
62G05 Nonparametric estimation


Full Text: DOI


[1] Box, G. E.P.; Jenkins, G. M.; Reinsel, G. C., Time Series Analysis Forecasting and Control (1994), Prentice-Hall Inc.: Prentice-Hall Inc. London · Zbl 0858.62072
[2] Cai, Z.; Fan, J.; Yao, Q., Function-coefficient regression models for nonlinear time series, J. Amer. Statist. Assoc., 95, 941-956 (2000) · Zbl 0996.62078
[3] Chen, H., Convergence rate for parametric components in a partly linear model, Ann. Statist., 16, 136-146 (1988) · Zbl 0637.62067
[4] Chen, R.; Tsay, S., Function-coefficient autoregressive models, J. Amer. Statist. Assoc., 88, 298-308 (1993) · Zbl 0776.62066
[5] Chiang, C.; Rica, J.; Wu, C., Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables, J. Amer. Statist. Assoc., 96, 605-619 (2001) · Zbl 1018.62034
[6] Engle, C. F.; Granger, W. J.; Rice, J.; Weiss, A., Semiparametric estimates of relation between weather and electricity sales, J. Amer. Statist. Assoc., 81, 310-320 (1986)
[7] Eubank, R. L., Spline Smoothing and Nonparametric Regression (1988), Marcel Dekker: Marcel Dekker New York · Zbl 0702.62036
[8] Fan, J.; Gijbels, I., Local Polynomial Modelling and its Applications (1996), Chapman & Hall: Chapman & Hall London · Zbl 0873.62037
[9] Fan, J.; Yao, Q., Nonlinear Time Series Nonparametric and Parametric Methods (2003), Springer: Springer New York · Zbl 1014.62103
[10] Fan, J.; Zhang, J., Statistical estimation in varying coefficient models, Ann. Statist., 27, 1491-1518 (1999) · Zbl 0977.62039
[11] Fan, J.; Zhang, J., Two-step estimation of functional linear models with applications to longitudinal data, R. Statist. Soc. B, 62, 303-322 (2000)
[12] Green, P. J.; Silverman, B. W., Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach (1994), Chapman & Hall: Chapman & Hall London · Zbl 0832.62032
[13] Hastie, T.; Tibshirani, R., Generalized Additive Models (1990), Chapman & Hall: Chapman & Hall London · Zbl 0747.62061
[14] Hastie, T.; Tibshirani, R., Varying-coefficient models, Roy. Statist. Soc., 55, 757-796 (1993) · Zbl 0796.62060
[15] He, X. M.; Shi, P. D., Bivariate tensor-product B-splines in partly linear model, J. Multivariate Anal., 58, 162-181 (1996) · Zbl 0865.62027
[16] Huang, J. Z.; Wu, C. O.; Zhou, L., Varying-coefficient models and basis function approximations for the analysis of repeated measurements, Biometrika, 89, 111-128 (2002) · Zbl 0998.62024
[17] Kotnour, K. D.; Box, G. E.P.; Altpeter, R. J., A discrete predictor-controller applied to sinusoidal perturbation adaptive optimization, Inst. Soc. Amer. Trans., 5, 255-262 (1966)
[18] Nadaraya, E. A., Nonparametric Estimation of Probability Densities and Regression Curves (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, (Translated by S. Kotz) · Zbl 0709.62039
[19] Speckman, P., Kernel smoothing in partial linear models, J. Roy. Statist. Soc. B, 50, 413-436 (1988) · Zbl 0671.62045
[20] H. Tong, Threshold Models in Non-linear Time Series Analysis, Lecture Notes in Statistics, vol. 21, Springer, Heidelberg, 1983.; H. Tong, Threshold Models in Non-linear Time Series Analysis, Lecture Notes in Statistics, vol. 21, Springer, Heidelberg, 1983. · Zbl 0527.62083
[21] Tong, H., Non-linear Time Series: A Dynamical System Approach (1990), Oxford University Press: Oxford University Press New York · Zbl 0716.62085
[22] Wand, M. P.; Jones, M. C., Kernel Smoothing (1995), Chapman & Hall: Chapman & Hall London · Zbl 0854.62043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.