Abstract
This article considers a semiparametric varying-coefficient partially linear regression model. The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable. A sieve M-estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. Our main object is to estimate the nonparametric component and the unknown parameters simultaneously. It is easier to compute and the required computation burden is much less than the existing two-stage estimation method. Furthermore, the sieve M-estimation is robust in the presence of outliers if we choose appropriate ρ(·). Under some mild conditions, the estimators are shown to be strongly consistent; the convergence rate of the estimator for the unknown nonparametric component is obtained and the estimator for the unknown parameter is shown to be asymptotically normally distributed. Numerical experiments are carried out to investigate the performance of the proposed method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ai C, Chen X. Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica, 2003, 71: 1795–1843
Bickel P J, Klaassen C A J, Ritov Y, et al. Efficient and Adaptive Inference for Semiparametric Models. Baltimore, MD: Johns Hopkins University Press, 1993
Boente G, He X, Zhou J. Robust estimates in generalized partially linear models. Ann Statist, 2006, 34: 2856–2878
Cai Z, Fan J, Li R. Efficient estimation and inferences for varying-coefficient models. J Amer Statist Assoc, 2000, 95: 888–902
Carroll R J, Fan J, Gijbels I, et al. Generalized partially linear single-Index models. J Amer Statist Assoc, 1997, 92: 477–489
Chamberlain G. Efficiency bounds for semiparametric regression. Econometrica, 1992, 60: 567–596
Chen G, You J. Wild bootstrap estimation in partially linear models with heteroscedasticity. Statist Probab Lett, 2006, 76: 340–348
Engle R F, Granger W J, Rice J, et al. Semiparametric estimates of the relation between weather and electricity sales. J Amer Statist Assoc, 1986, 80: 310–319
Fan J, Huang T. Profile likelihood inferences on semiparametric varying coefficient partially linear models. Bernoulli, 2005, 11: 1031–1057
Fan J, Zhang W. Statistical estimation in varying coefficient models. Ann Statist, 1999, 27: 1491–1518
Hastie T J, Tibshirani R. Varying-coefficient models. J Roy Statist Soc B, 1993, 55: 757–96
Härdle W, Liang H, Gao J. Partially Linear Models. Heidelberg: Physica-Verlag, 2000
He X, Shi P. Bivariate tensor-product B-splines in a partly linear model. J Multivariate Anal, 1996, 58: 162–181
Huber P. Robust Statistics. New York: John Wiley, 1981
Hurvich C M, Tsai C L. Regression and time series model selection in small samples. Biometrika, 1989, 76: 297–307
Ahmad I, Leelahanon S, Li Q. Efficient estimation of a semiparametric partially linear varying coefficient model. Ann Statist, 2005, 33: 258–283
Li Q, Huang C J, Li D, et al. Semiparametric smooth coefficient models. J Bus Econ Statist, 2002, 3: 412–422
Li R, Liang H. Variable selection in semiparametric regression modeling. Ann Statist, 2008, 36: 261–286
Maronna R A, Martin D R, Yohai V J. Robust Statistics: Theory and Methods. London: Wiley, 2006
Ossiander M. A central limit theorem under metric entropy with L 2 bracketing. Ann Probab, 1987, 15: 897–919
Pollard D. Convergence of Stochastic Processes. New York: Springer, 1984
Ruppert D, Wand M, Carroll R. Semiparametric Regression. Cambridge: Cambridge University Press, 2003
Schumaker L. Spline Functions: Basic Theory. New York: John Wiley, 1981
Shen X. On methods of sieves and penalization. Ann Statist, 1997, 25: 2555–2591
Shen X, Wong W H. Convergence rate of sieve estimates. Ann Statist, 1994, 22: 580–615
Stone C J. Optimal global rates of convergence for nonparametric regression. Ann Statist, 1982, 10: 1040–1053
van de Geer S A. Empirical Processes in M-Estimation. Cambridge: Cambridge University Press, 2000
van der Vaart A W, Wellner J. Weak Convergence and Empirical Processes, with Applications to Statistics. New York: Springer-Verlag, 1996
Wellner J, Zhang Y. Two likelihood-based semiparametric estimation methods for panel count data with covariates. Ann Statist, 2007, 35: 2106–2142
Wheeden R L, Zygmund A. Measure and Integral. New York: M. Dekker, 1977
Xue H, Lam K F, Li G. Sieve maximum likelihood estimation for semiparametric regression models with current status data. J Amer Statist Assoc, 2004, 99: 346–356
Yatchew A. Semiparametric Regression for the Applied Econometrician. Cambridge: Cambridge University Press, 2003
You J, Chen G. Estimation of a semiparametric varying coefficient partially linear errors in variables model. J Multivariate Anal, 2006, 97: 325–341
You J, Zhou Y. Empirical likelihood for semiparametric varying coefficient partially linear regression models. Statist Probab Lett, 2006, 11: 412–422
Zhang W, Lee S, Song X. Local polynomial fitting in semivarying coefficient models. J Multivariate Anal, 2002, 82: 166–188
Zhou X, You J. Wavelet estimation in varying-coefficient partially linear regression models. Statist Probab Lett, 2004, 68: 91–104
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, T., Cui, H. Sieve M-estimation for semiparametric varying-coefficient partially linear regression model. Sci. China Math. 53, 1995–2010 (2010). https://doi.org/10.1007/s11425-010-4030-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-4030-7