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A robust and efficient estimation method for single index models

Author

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  • Liu, Jicai
  • Zhang, Riquan
  • Zhao, Weihua
  • Lv, Yazhao

Abstract

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They have applications to many fields, such as medicine, economics and finance. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. Although an M-type regression is often considered as a good alternative to those methods, it may lose efficiency for normal errors. In this paper, we propose a new robust and efficient estimation procedure based on local modal regression for single index models. The asymptotic normality of proposed estimators for both the parametric and nonparametric parts is established. We show that the proposed estimators are as asymptotically efficient as the least-square-based estimators when there are no outliers and the error distribution is normal. A modified EM algorithm is presented for efficient implementation. The simulations and real data analysis are conducted to illustrate the finite sample performance of the proposed method.

Suggested Citation

  • Liu, Jicai & Zhang, Riquan & Zhao, Weihua & Lv, Yazhao, 2013. "A robust and efficient estimation method for single index models," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 226-238.
    • Handle: RePEc:eee:jmvana:v:122:y:2013:i:c:p:226-238
    DOI: 10.1016/j.jmva.2013.08.007
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    References listed on IDEAS

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    1. Lee, Myoung-jae, 1989. "Mode regression," Journal of Econometrics, Elsevier, vol. 42(3), pages 337-349, November.
    2. Wu, Tracy Z. & Yu, Keming & Yu, Yan, 2010. "Single-index quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1607-1621, August.
    3. 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.
    4. 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.
    5. 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, August.
    6. Lee, Myoung-jae, 1993. "Quadratic mode regression," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 1-19.
    7. Th. Gasser & P. Hall & B. Presnell, 1998. "Nonparametric estimation of the mode of a distribution of random curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 681-691.
    8. Weixin Yao & Bruce Lindsay & Runze Li, 2012. "Local modal regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(3), pages 647-663.
    9. Lai, Peng & Wang, Qihua & Lian, Heng, 2012. "Bias-corrected GEE estimation and smooth-threshold GEE variable selection for single-index models with clustered data," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 422-432.
    10. Xueqin Wang & Yunlu Jiang & Mian Huang & Heping Zhang, 2013. "Robust Variable Selection With Exponential Squared Loss," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 632-643, June.
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    Citations

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    Cited by:

    1. Wang, Kangning & Li, Shaomin & Sun, Xiaofei & Lin, Lu, 2019. "Modal regression statistical inference for longitudinal data semivarying coefficient models: Generalized estimating equations, empirical likelihood and variable selection," Computational Statistics & Data Analysis, Elsevier, vol. 133(C), pages 257-276.
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    3. Chaohui Guo & Hu Yang & Jing Lv, 2018. "Two step estimations for a single-index varying-coefficient model with longitudinal data," Statistical Papers, Springer, vol. 59(3), pages 957-983, September.
    4. Jiang, Rong & Qian, Wei-Min & Zhou, Zhan-Gong, 2016. "Weighted composite quantile regression for single-index models," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 34-48.
    5. Claudio Agostinelli & Ana M. Bianco & Graciela Boente, 2020. "Robust estimation in single-index models when the errors have a unimodal density with unknown nuisance parameter," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 855-893, June.
    6. Kangning Wang & Lu Lin, 2019. "Robust and efficient estimator for simultaneous model structure identification and variable selection in generalized partial linear varying coefficient models with longitudinal data," Statistical Papers, Springer, vol. 60(5), pages 1649-1676, October.
    7. Huybrechts F. Bindele & Ash Abebe & Karlene N. Meyer, 2018. "General rank-based estimation for regression single index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1115-1146, October.
    8. Ash Abebe & Huybrechts F. Bindele & Masego Otlaadisa & Boikanyo Makubate, 2021. "Robust estimation of single index models with responses missing at random," Statistical Papers, Springer, vol. 62(5), pages 2195-2225, October.
    9. Yang, Hu & Guo, Chaohui & Lv, Jing, 2014. "A robust and efficient estimation method for single-index varying-coefficient models," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 119-127.
    10. Lv, Zhike & Zhu, Huiming & Yu, Keming, 2014. "Robust variable selection for nonlinear models with diverging number of parameters," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 90-97.
    11. Yunlu Jiang & Guo-Liang Tian & Yu Fei, 2019. "A robust and efficient estimation method for partially nonlinear models via a new MM algorithm," Statistical Papers, Springer, vol. 60(6), pages 2063-2085, December.
    12. Yang, Jing & Tian, Guoliang & Lu, Fang & Lu, Xuewen, 2020. "Single-index modal regression via outer product gradients," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    13. Yang, Hu & Yang, Jing, 2014. "A robust and efficient estimation and variable selection method for partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 227-242.
    14. Qingyang Liu & Xianzheng Huang & Haiming Zhou, 2024. "The Flexible Gumbel Distribution: A New Model for Inference about the Mode," Stats, MDPI, vol. 7(1), pages 1-16, March.
    15. Han, Zhong-Cheng & Lin, Jin-Guan & Zhao, Yan-Yong, 2020. "Adaptive semiparametric estimation for single index models with jumps," Computational Statistics & Data Analysis, Elsevier, vol. 151(C).
    16. Lv, Jing & Yang, Hu & Guo, Chaohui, 2015. "An efficient and robust variable selection method for longitudinal generalized linear models," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 74-88.
    17. Han, Jinyue & Wang, Jun & Gao, Wei & Tang, Man-Lai, 2023. "Estimation of the directions for unknown parameters in semiparametric models," MPRA Paper 116365, University Library of Munich, Germany.

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