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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.

MSC:

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

Citations:

Zbl 1029.62042

Software:

KernSmooth
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Full Text: DOI Euclid

References:

[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 · doi:10.2307/2669837
[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 · doi:10.2307/2669472
[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 · doi:10.2307/2965697
[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 · doi:10.2307/2669618
[7] Chamberlain, G. (1992) Efficient bounds for semiparametric regression. Econometrika, 60, 567-596. JSTOR: · Zbl 0774.62038 · doi:10.2307/2951584
[8] Chen, R. and Tsay, R.J. (1993) Functional-coefficient autoregressive models. J. Amer. Statist. Assoc., 88, 298-308. JSTOR: · Zbl 0776.62066 · doi:10.2307/2290725
[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 · doi:10.1198/016214501753168316
[14] Fan, J. and Zhang, W. (1999) Statistical estimation in varying coefficient models. Ann. Statist., 27, 1491-1518. · Zbl 0977.62039 · doi:10.1214/aos/1017939139
[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 · doi:10.1111/1467-9469.00218
[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 · doi:10.1214/aos/996986505
[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 · doi:10.1093/biomet/68.1.189
[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 · doi:10.2307/2670060
[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 · doi:10.1017/S026646660420202X
[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 · doi:10.1016/0095-0696(78)90006-2
[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 · doi:10.1093/biomet/85.4.809
[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 · doi:10.1093/biomet/89.1.111
[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 · doi:10.1214/aos/1017939140
[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 · doi:10.1007/BF00539840
[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 · doi:10.2307/2965570
[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 · doi:10.2307/2291516
[33] Severini, T.A. and Wong, W.H. (1992) Generalized profile likelihood and conditional parametric models. Ann. Statist., 20, 1768-1802. · Zbl 0768.62015 · doi:10.1214/aos/1176348889
[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 · doi:10.1016/S0165-1765(97)00218-8
[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 · doi:10.1006/jmva.2001.2012
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