Bernoulli

  • Bernoulli
  • Volume 11, Number 6 (2005), 1031-1057.

Profile likelihood inferences on semiparametric varying-coefficient partially linear models

Jianqing Fan and Tao Huang

Full-text: Open access

Abstract

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 Fan et al. 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 χ2 distribution under the null hypothesis. This not only unveils a new Wilks type of 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.

Article information

Source
Bernoulli Volume 11, Number 6 (2005), 1031-1057.

Dates
First available in Project Euclid: 16 January 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1137421639

Digital Object Identifier
doi:10.3150/bj/1137421639

Mathematical Reviews number (MathSciNet)
MR2189080

Zentralblatt MATH identifier
1098.62077

Keywords
generalized likelihood ratio statistics local linear regression partially linear models profile likelihood varying-coefficient partially linear models Wald statistics

Citation

Fan, Jianqing; Huang, Tao. Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11 (2005), no. 6, 1031--1057. doi:10.3150/bj/1137421639. https://projecteuclid.org/euclid.bj/1137421639


Export citation

References

  • [1] Bickel, P.J. and Kwon, J. (2001) Inference for semiparametric models: Some current frontiers (with discussion). Statist. Sinica, 11, 863-960.
  • [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.
  • [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.
  • [4] Cai, Z., Fan, J. and Li, R. (2000) Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc., 95, 888-902.
  • [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.
  • [6] Carroll, R.J., Ruppert, D. and Welsh, A.H. (1998) Nonparametric estimation via local estimating equations. J. Amer. Statist. Assoc., 93, 214-227.
  • [7] Chamberlain, G. (1992) Efficient bounds for semiparametric regression. Econometrika, 60, 567-596.
  • [8] Chen, R. and Tsay, R.J. (1993) Functional-coefficient autoregressive models. J. Amer. Statist. Assoc., 88, 298-308.
  • [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.
  • [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.
  • [12] Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. London: Chapman & Hall.
  • [13] Fan, J. and Huang, L. (2001) Goodness-of-fit test for parametric regression models. J. Amer. Statist. Assoc., 96, 640-652.
  • [14] Fan, J. and Zhang, W. (1999) Statistical estimation in varying coefficient models. Ann. Statist., 27, 1491-1518.
  • [15] Fan, J. and Zhang, W. (2000) Simultaneous confidence bands and hypothesis testing in varying coefficient models. Scand. J. Statist., 27, 715-731.
  • [16] Fan, J., Zhang, C. and Zhang, J. (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Ann. of Statist., 29, 153-193.
  • [17] Green, P.J. and Silverman, B.W. (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. London: Chapman & Hall.
  • [18] Haggan, V. and Ozaki, T. (1981) Modeling nonlinear vibrations using an amplitude-dependent autoregressive time series model. Biometrika, 68, 189-196.
  • [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.
  • [20] Härdle, W., Liang, H. and Gao, J.T. (2000) Partially Linear Models. New York: Springer-Verlag.
  • [21] Härdle, W., Huet, S., Mammen, E. and Sperlich, S. (2004) Bootstrap inference in semiparametric generalized additive models. Econometric Theory, 20, 265-300.
  • [22] Harrison, D. and Rubinfeld, D.L. (1978) Hedonic prices and the demand for clean air. J. Environ. Economics Management, 5, 81-102.
  • [23] Hastie, T.J. and Tibshirani, R. (1990) Generalized Additive Models. London: Chapman & Hall.
  • [24] Hastie, T.J. and Tibshirani, R. (1993) Varying-coefficient models. J. Roy. Statist. Soc. Ser. B, 55, 757-796.
  • [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.
  • [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.
  • [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.
  • [28] Li, Q., Huang, C.J., Li, D. and Fu, T.T. (2002) Semiparametric smooth coefficient models. J. Business Econom. Statist., 20, 412-422.
  • [29] Liang, H., Härdle, W. and Carroll, R.J. (1999) Estimation in a semiparametric partially linear errorsin- variables model. Ann. Statist., 27, 1519-1535.
  • [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.
  • [31] Ruppert, D. (1997) Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. J. Amer. Statist. Assoc., 92, 1049-1062.
  • [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.
  • [33] Severini, T.A. and Wong, W.H. (1992) Generalized profile likelihood and conditional parametric models. Ann. Statist., 20, 1768-1802.
  • [34] Speckman, P. (1988) Kernel smoothing in partial linear models. J. Roy. Statist. Soc. B, 50, 413-436.
  • [35] Wand, M.P. and Jones, M.C. (1995) Kernel Smoothing. London: Chapman & Hall.
  • [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.
  • [38] Yatchew, A. (1997) An elementary estimator for the partial linear model. Economics Lett., 57, 135-143.
  • [39] Zhang, W., Lee, S.-Y. and Song, X. (2002) Local polynomial fitting in semivarying coefficient models. J. Multivariate Anal., 82, 166-188.