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# M-estimators for single-index model using B-spline

## Author Info

Listed author(s):
• Qingming Zou
• Zhongyi Zhu

()

Registered author(s):

## Abstract

The single-index model is an important tool in multivariate nonparametric regression. This paper deals with M-estimators for the single-index model. Unlike the existing M-estimator for the single-index model, the unknown link function is approximated by B-spline and M-estimators for the parameter and the nonparametric component are obtained in one step. The proposed M-estimator of unknown function is shown to attain the convergence rate as that of the optimal global rate of convergence of estimators for nonparametric regression according to Stone (Ann Stat 8:1348–1360, 1980 ; Ann Stat 10:1040–1053, 1982 ), and the M-estimator of parameter is $$\sqrt{n}$$ -consistent and asymptotically normal. A small sample simulation study showed that the M-estimators proposed in this paper are robust. An application to real data illustrates the estimator’s usefulness. Copyright Springer-Verlag Berlin Heidelberg 2014

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File URL: http://hdl.handle.net/10.1007/s00184-013-0434-z
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## Bibliographic Info

Article provided by Springer in its journal Metrika.

Volume (Year): 77 (2014)
Issue (Month): 2 (February)
Pages: 225-246

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 Handle: RePEc:spr:metrik:v:77:y:2014:i:2:p:225-246 DOI: 10.1007/s00184-013-0434-z Contact details of provider: Web page: http://www.springer.com Order Information: Web: http://www.springer.com/statistics/journal/184/PS2

## References

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1. Prasad Naik & Chih-Ling Tsai, 2000. "Partial least squares estimator for single-index models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 763-771.
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3. Yingcun Xia & Howell Tong & W. K. Li & Li-Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410.
4. Jiti Gao & Hua Liang, 1997. "Statistical Inference in Single-Index and Partially Nonlinear Models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(3), pages 493-517, September.
5. Stoker, Thomas M, 1986. "Consistent Estimation of Scaled Coefficients," Econometrica, Econometric Society, vol. 54(6), pages 1461-1481, November.
6. Wu, Tracy Z. & Yu, Keming & Yu, Yan, 2010. "Single-index quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1607-1621, August.
7. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
8. Xia, Yingcun & Härdle, Wolfgang, 2006. "Semi-parametric estimation of partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1162-1184, May.
9. Jianhua Z. Huang, 2002. "Varying-coefficient models and basis function approximations for the analysis of repeated measurements," Biometrika, Biometrika Trust, vol. 89(1), pages 111-128, March.
10. Li, Jianbo & Zhang, Riquan, 2011. "Partially varying coefficient single index proportional hazards regression models," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 389-400, January.
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