Profile likelihood inferences on semiparametric varying-coefficient partially linear models. (English) Zbl 1098.62077

Summary: Varying-coefficient partially linear models are frequently used in statistical modelling, but their estimation and inference have not been systematically studied. This paper proposes a profile least-squares technique for estimating the parametric component and studies the asymptotic normality of the profile least-squares estimator. The main focus is the examination of whether the generalized likelihood technique developed by J. Fan et al. [Ann. Stat. 29, No. 1, 153–193 (2001; Zbl 1029.62042)] is applicable to the testing problem for the parametric component of semiparametric models.
We introduce the profile likelihood ratio test and demonstrate that it follows an asymptotically \(\chi^2\) distribution under the null hypothesis. This not only unveils a new Wilks type phenomenon, but also provides a simple and useful method for semiparametric inferences. In addition, the Wald statistic for semiparametric models is introduced and demonstrated to possess a sampling property similar to the profile likelihood ratio statistic. A new and simple bandwidth selection technique is proposed for semiparametric inferences on partially linear models and numerical examples are presented to illustrate the proposed methods.


62H15 Hypothesis testing in multivariate analysis
62J05 Linear regression; mixed models
62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics
62H10 Multivariate distribution of statistics


Zbl 1029.62042


Full Text: DOI Euclid


[1] Bickel, P.J. and Kwon, J. (2001) Inference for semiparametric models: Some current frontiers (with discussion). Statist. Sinica, 11, 863-960. · Zbl 0997.62028
[2] Bickel, P.J., Klaassen, A.J., Ritov, Y. and Wellner, J.A. (1993) Efficient and Adaptive Inference in Semi-parametric Models. Baltimore, MD: Johns Hopkins University Press. · Zbl 0786.62001
[3] Brumback, B. and Rice, J.A. (1998) Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion). J. Amer. Statist. Assoc., 93, 961-994. JSTOR: · Zbl 1064.62515
[4] Cai, Z., Fan, J. and Li, R. (2000) Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc., 95, 888-902. JSTOR: · Zbl 0999.62052
[5] Carroll, R.J., Fan, J., Gijbels, I, and Wand, M.P. (1997) Generalized partially linear single-index models. J. Amer. Satist. Assoc., 92, 477-489. JSTOR: · Zbl 0890.62053
[6] Carroll, R.J., Ruppert, D. and Welsh, A.H. (1998) Nonparametric estimation via local estimating equations. J. Amer. Statist. Assoc., 93, 214-227. JSTOR: · Zbl 0910.62033
[7] Chamberlain, G. (1992) Efficient bounds for semiparametric regression. Econometrika, 60, 567-596. JSTOR: · Zbl 0774.62038
[8] Chen, R. and Tsay, R.J. (1993) Functional-coefficient autoregressive models. J. Amer. Statist. Assoc., 88, 298-308. JSTOR: · Zbl 0776.62066
[9] Cleveland, W.S., Grosse, E. and Shyu, W.M. (1991) Local regression models. In J.M. Chambers and T.J. Hastie (eds.), Statistical Models in S, pp. 309-376. Pacific Grove, CA: Wadsworth/Brooks-Cole.
[10] Cuzick, J. (1992) Semiparametric additive regression. J. Roy. Statist. Soc. Ser. B, 54, 831-843. JSTOR: · Zbl 0776.62036
[11] Fan, J. and Gijbels, I. (1995) Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaption. J. Roy. Statist. Soc. Ser. B, 57, 371-394. JSTOR: · Zbl 0813.62033
[12] Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. London: Chapman & Hall. · Zbl 0873.62037
[13] Fan, J. and Huang, L. (2001) Goodness-of-fit test for parametric regression models. J. Amer. Statist. Assoc., 96, 640-652. JSTOR: · Zbl 1017.62014
[14] Fan, J. and Zhang, W. (1999) Statistical estimation in varying coefficient models. Ann. Statist., 27, 1491-1518. · Zbl 0977.62039
[15] Fan, J. and Zhang, W. (2000) Simultaneous confidence bands and hypothesis testing in varying coefficient models. Scand. J. Statist., 27, 715-731. · Zbl 0962.62032
[16] Fan, J., Zhang, C. and Zhang, J. (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Ann. of Statist., 29, 153-193. · Zbl 1029.62042
[17] Green, P.J. and Silverman, B.W. (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. London: Chapman & Hall. · Zbl 0832.62032
[18] Haggan, V. and Ozaki, T. (1981) Modeling nonlinear vibrations using an amplitude-dependent autoregressive time series model. Biometrika, 68, 189-196. JSTOR: · Zbl 0462.62070
[19] Härdle, W., Mammen, E. and Müller, M. (1998) Testing parametric versus semiparametric modelling in generalized linear models. J. Amer. Statist. Assoc., 93, 1461-1474. · Zbl 1064.62543
[20] Härdle, W., Liang, H. and Gao, J.T. (2000) Partially Linear Models. New York: Springer-Verlag. · Zbl 0968.62006
[21] Härdle, W., Huet, S., Mammen, E. and Sperlich, S. (2004) Bootstrap inference in semiparametric generalized additive models. Econometric Theory, 20, 265-300. · Zbl 1072.62034
[22] Harrison, D. and Rubinfeld, D.L. (1978) Hedonic prices and the demand for clean air. J. Environ. Economics Management, 5, 81-102. · Zbl 0375.90023
[23] Hastie, T.J. and Tibshirani, R. (1990) Generalized Additive Models. London: Chapman & Hall. · Zbl 0747.62061
[24] Hastie, T.J. and Tibshirani, R. (1993) Varying-coefficient models. J. Roy. Statist. Soc. Ser. B, 55, 757-796. JSTOR: · Zbl 0796.62060
[25] Hoover, D.R., Rice, J.A., Wu, C.O. and Yang, L.-P. (1998) Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika, 85, 809-822. JSTOR: · Zbl 0921.62045
[26] Huang, J.Z., Wu, C.O. and Zhou, L. (2002) Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika, 89, 111-128. JSTOR: · Zbl 0998.62024
[27] Ingster, Yu.I. (1993) Asymptotically minimax hypothesis testing for nonparametric alternatives I-III. Math. Methods Statist., 2, 85-114; 3, 171-189; 4, 249-268. · Zbl 0798.62059
[28] Li, Q., Huang, C.J., Li, D. and Fu, T.T. (2002) Semiparametric smooth coefficient models. J. Business Econom. Statist., 20, 412-422. JSTOR:
[29] Liang, H., Härdle, W. and Carroll, R.J. (1999) Estimation in a semiparametric partially linear errorsin- variables model. Ann. Statist., 27, 1519-1535. · Zbl 0977.62036
[30] Mack, Y.P. and Silverman, B.W. (1982) Weak and strong uniform consistency of kernel regression estimates. Z. Wahrscheinlichkeitstheorie Verw. Geb., 61, 405-415. · Zbl 0495.62046
[31] Ruppert, D. (1997) Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. J. Amer. Statist. Assoc., 92, 1049-1062. JSTOR: · Zbl 1067.62531
[32] Ruppert, D., Sheathers, S.J. and Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. J. Amer. Statist. Assoc., 90, 1257-1270. JSTOR: · Zbl 0868.62034
[33] Severini, T.A. and Wong, W.H. (1992) Generalized profile likelihood and conditional parametric models. Ann. Statist., 20, 1768-1802. · Zbl 0768.62015
[34] Speckman, P. (1988) Kernel smoothing in partial linear models. J. Roy. Statist. Soc. B, 50, 413-436. JSTOR: · Zbl 0671.62045
[35] Wand, M.P. and Jones, M.C. (1995) Kernel Smoothing. London: Chapman & Hall. · Zbl 0854.62043
[36] Wahba, G. (1984) Partial spline models for semiparametric estimation of functions of several variables. In Statistical Analysis of Time Series, Proceedings of the Japan-U.S. Joint Seminar, Tokyo, pp. 319-329. Tokyo Institute of Statistical Mathematics.
[37] Xia, Y. and Li, W.K. (1999) On the estimation and testing of functional-coefficient linear models. Statist. Sinica, 9, 735-757. · Zbl 0958.62040
[38] Yatchew, A. (1997) An elementary estimator for the partial linear model. Economics Lett., 57, 135-143. · Zbl 0896.90052
[39] Zhang, W., Lee, S.-Y. and Song, X. (2002) Local polynomial fitting in semivarying coefficient models. J. Multivariate Anal., 82, 166-188. · Zbl 0995.62038
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.