Estimation and testing for partially linear single-index models
Hua Liang, Xiang Liu, Runze Li, and Chih-Ling Tsai
Source: Ann. Statist. Volume 38, Number 6
(2010), 3811-3836.
Abstract
In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.
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Keywords: Efficiency; hypothesis testing; local linear regression; nonparametric regression; profile likelihood; SCAD
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1291126974
Digital Object Identifier: doi:10.1214/10-AOS835
Zentralblatt MATH identifier: 1204.62068
Mathematical Reviews number (MathSciNet): MR2766869
References
Akaike, H. (1973). Information theory as an extension of the maximum likelihood principle. In 2nd International Symposium on Information Theory (B. N. Petrov and F. Csaki, eds.) 267–281. Akademia Kiado, Budapest.
Mathematical Reviews (MathSciNet): MR483125
Zentralblatt MATH: 0283.62006
Bickel, P. J., Klaasen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD.
Mathematical Reviews (MathSciNet): MR1245941
Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477–489.
Mathematical Reviews (MathSciNet): MR1467842
Zentralblatt MATH: 0890.62053
Digital Object Identifier: doi:10.2307/2965697
JSTOR: links.jstor.org
Duan, N. H. and Li, K. C. (1991). Slicing regression: A link-free regression method. Ann. Statist. 19 505–530.
Mathematical Reviews (MathSciNet): MR1105834
Zentralblatt MATH: 0738.62070
Digital Object Identifier: doi:10.1214/aos/1176348109
Project Euclid: euclid.aos/1176348109
Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall, New York.
Mathematical Reviews (MathSciNet): MR1383587
Zentralblatt MATH: 0873.62037
Fan, J. and Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11 1031–1057.
Mathematical Reviews (MathSciNet): MR2189080
Digital Object Identifier: doi:10.3150/bj/1137421639
Project Euclid: euclid.bj/1137421639
Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
Mathematical Reviews (MathSciNet): MR1946581
Zentralblatt MATH: 1073.62547
Digital Object Identifier: doi:10.1198/016214501753382273
JSTOR: links.jstor.org
Fan, J. and Li, R. (2004). New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. J. Amer. Statist. Assoc. 99 710–723.
Mathematical Reviews (MathSciNet): MR2090905
Zentralblatt MATH: 1117.62329
Digital Object Identifier: doi:10.1198/016214504000001060
Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistical and Wilks phenomenon. Ann. Statist. 29 153–193.
Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178.
Mathematical Reviews (MathSciNet): MR1212171
Zentralblatt MATH: 0770.62049
Digital Object Identifier: doi:10.1214/aos/1176349020
Project Euclid: euclid.aos/1176349020
Härdle, W., Liang, H. and Gao, J. (2000). Partially Linear Models. Springer Physica-Verlag, Heidelberg.
Mathematical Reviews (MathSciNet): MR1787637
Horowitz, J. (1998). Semiparametric Methods in Econometrics. Lecture Notes in Statistics. Springer, New York.
Mathematical Reviews (MathSciNet): MR1624936
Horowitz, J. L. and Härdle, W. (1996). Direct semiparametric estimation of single-index models with discrete covariates. J. Amer. Statist. Assoc. 91 1632–1639.
Mathematical Reviews (MathSciNet): MR1439104
Digital Object Identifier: doi:10.2307/2291590
JSTOR: links.jstor.org
Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120.
Mathematical Reviews (MathSciNet): MR1230981
Zentralblatt MATH: 0816.62079
Digital Object Identifier: doi:10.1016/0304-4076(93)90114-K
Jennrich, R. I. (1969). Asymptotic properties of nonlinear least squares estimators. Ann. Math. Statist. 40 633–643.
Mathematical Reviews (MathSciNet): MR238419
Zentralblatt MATH: 0193.47201
Digital Object Identifier: doi:10.1214/aoms/1177697731
Project Euclid: euclid.aoms/1177697731
Kong, E. and Xia, Y. C. (2007). Variable selection for the single-index model. Biometrika 94 217–229.
Mathematical Reviews (MathSciNet): MR2367831
Zentralblatt MATH: 1142.62353
Digital Object Identifier: doi:10.1093/biomet/asm008
Liang, H. and Wang, N. (2005). Partially linear single-index measurement error models. Statist. Sinica 15 99–116.
Mathematical Reviews (MathSciNet): MR2125722
Zentralblatt MATH: 1061.62098
Mack, Y. and Silverman, B. (1982). Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 60 405–415.
Mathematical Reviews (MathSciNet): MR679685
Zentralblatt MATH: 0881.62037
Newey, W. K. (1994). The asymptotic variance of semiparametric estimators. Econometrica 62 1349–1382.
Mathematical Reviews (MathSciNet): MR1303237
Digital Object Identifier: doi:10.2307/2951752
JSTOR: links.jstor.org
Powell, J. L., Stock, J. H. and Stoker, T. M. (1989). Semiparametric estimation of index coefficient. Econometrica 51 1403–1430.
Mathematical Reviews (MathSciNet): MR1035117
Digital Object Identifier: doi:10.2307/1913713
JSTOR: links.jstor.org
Seifert, B. and Gasser, T. (1996). Finite-sample variance of local polynomials: Analysis and solutions. J. Amer. Statist. Assoc. 91 267–275.
Mathematical Reviews (MathSciNet): MR1394081
Zentralblatt MATH: 0871.62042
Digital Object Identifier: doi:10.2307/2291404
JSTOR: links.jstor.org
Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768–1802.
Mathematical Reviews (MathSciNet): MR1193312
Zentralblatt MATH: 0768.62015
Digital Object Identifier: doi:10.1214/aos/1176348889
Project Euclid: euclid.aos/1176348889
Speckman, P. (1988). Kernel smoothing in partial linear models. J. R. Stat. Soc. Ser. B Stat. Methodol. 50 413–436.
Mathematical Reviews (MathSciNet): MR970977
JSTOR: links.jstor.org
Tibshirani, R. (1996). Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. Ser. B Stat. Methodol. 58 267-288.
Mathematical Reviews (MathSciNet): MR1379242
JSTOR: links.jstor.org
Wang, H. S., Li, R. and Tsai, C. L. (2007). Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika 94 553–568.
Mathematical Reviews (MathSciNet): MR2410008
Zentralblatt MATH: 1135.62058
Digital Object Identifier: doi:10.1093/biomet/asm053
Xia, Y. C. and Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models. J. Multivariate Anal. 97 1162–1184.
Mathematical Reviews (MathSciNet): MR2276153
Zentralblatt MATH: 1089.62050
Digital Object Identifier: doi:10.1016/j.jmva.2005.11.005
Xia, Y. C., Tong, H., Li, W. K. and Zhu, L. (2002). An adaptive estimation of dimension reduction space (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 64 363–410.
Mathematical Reviews (MathSciNet): MR1924297
Zentralblatt MATH: 1091.62028
Digital Object Identifier: doi:10.1111/1467-9868.03411
JSTOR: links.jstor.org
Yu, Y. and Ruppert, D. (2002). Penalized spline estimation for partially linear single-index models. J. Amer. Statist. Assoc. 97 1042–1054.
Mathematical Reviews (MathSciNet): MR1951258
Zentralblatt MATH: 1045.62035
Digital Object Identifier: doi:10.1198/016214502388618861
JSTOR: links.jstor.org
Zhang, Y., Li, R. and Tsai, C.-L. (2010). Regularization parameter selections via generalized information criterion. J. Amer. Statist. Assoc. 105 312–323.
Mathematical Reviews (MathSciNet): MR2656055
Digital Object Identifier: doi:10.1198/jasa.2009.tm08013
