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Robust variable selection for the varying coefficient model based on composite L 1 -- L 2 regression

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  • Weihua Zhao
  • Riquan Zhang
  • Jicai Liu

Abstract

The varying coefficient model (VCM) is an important generalization of the linear regression model and many existing estimation procedures for VCM were built on L 2 loss, which is popular for its mathematical beauty but is not robust to non-normal errors and outliers. In this paper, we address the problem of both robustness and efficiency of estimation and variable selection procedure based on the convex combined loss of L 1 and L 2 instead of only quadratic loss for VCM. By using local linear modeling method, the asymptotic normality of estimation is driven and a useful selection method is proposed for the weight of composite L 1 and L 2 . Then the variable selection procedure is given by combining local kernel smoothing with adaptive group LASSO. With appropriate selection of tuning parameters by Bayesian information criterion (BIC) the theoretical properties of the new procedure, including consistency in variable selection and the oracle property in estimation, are established. The finite sample performance of the new method is investigated through simulation studies and the analysis of body fat data. Numerical studies show that the new method is better than or at least as well as the least square-based method in terms of both robustness and efficiency for variable selection.

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  • Weihua Zhao & Riquan Zhang & Jicai Liu, 2013. "Robust variable selection for the varying coefficient model based on composite L 1 -- L 2 regression," Journal of Applied Statistics, Taylor & Francis Journals, vol. 40(9), pages 2024-2040, September.
    • Handle: RePEc:taf:japsta:v:40:y:2013:i:9:p:2024-2040
    DOI: 10.1080/02664763.2013.804040
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    Cited by:

    1. Ullah, Aman & Wang, Tao & Yao, Weixin, 2023. "Semiparametric partially linear varying coefficient modal regression," Journal of Econometrics, Elsevier, vol. 235(2), pages 1001-1026.
    2. Lu Li & Kai Tan & Xuerong Meggie Wen & Zhou Yu, 2023. "Variable-dependent partial dimension reduction," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 521-541, June.
    3. Jin-Guan Lin & Yan-Yong Zhao & Hong-Xia Wang, 2015. "Heteroscedasticity diagnostics in varying-coefficient partially linear regression models and applications in analyzing Boston housing data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(11), pages 2432-2448, November.
    4. Fang Lu & Shengjie Li & Jing Yang, 2015. "Convergence analysis of weighted expected residual method for nonlinear stochastic variational inequality problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(2), pages 229-242, October.
    5. 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.
    6. Zhao, Yan-Yong & Lin, Jin-Guan & Huang, Xing-Fang & Wang, Hong-Xia, 2016. "Adaptive jump-preserving estimates in varying-coefficient models," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 65-80.
    7. Germán Aneiros & Philippe Vieu, 2015. "Partial linear modelling with multi-functional covariates," Computational Statistics, Springer, vol. 30(3), pages 647-671, September.

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