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A robust and efficient estimation method for partially nonlinear models via a new MM algorithm

Author

Listed:
  • Yunlu Jiang

    (Jinan University)

  • Guo-Liang Tian

    (Southern University of Science and Technology)

  • Yu Fei

    (Yunnan University of Finance and Economics)

Abstract

When the observed data set contains outliers, it is well known that the classical least squares method is not robust. To overcome this difficulty, Wang et al. (J Am Stat Assoc 108(502): 632–643, 2013) proposed a robust variable selection method by using the exponential squared loss (ESL) function with a tuning parameter. Although many important statistical models are investigated, to date, in the presence of outliers there is no paper to study the partially nonlinear model by using the ESL function. To fill in this gap, in this paper, we propose a robust and efficient estimation method for the partially nonlinear model based on the ESL function. Under certain conditions, we have shown that the proposed estimators can achieve the best convergence rates. Next, the asymptotic normality of the proposed estimators is established. In addition, we develop a new minorization–maximization algorithm to calculate the estimates for both non-parametric and parametric parts and present a procedure for deriving initial values. Finally, we provide a data-driven approach to select the tuning parameters. Numerical simulations and a real data analysis are used to illustrate that when there are outliers, the proposed ESL method is more robust and efficient for partially nonlinear models than the existing linear approximation method and the composite quantile regression method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:60:y:2019:i:6:d:10.1007_s00362-017-0909-5
    DOI: 10.1007/s00362-017-0909-5
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    References listed on IDEAS

    as
    1. Yunlu Jiang, 2015. "Robust estimation in partially linear regression models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(11), pages 2497-2508, November.
    2. Huang, Tzee-Ming & Chen, Hung, 2008. "Estimating the parametric component of nonlinear partial spline model," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1665-1680, September.
    3. Weixin Yao & Bruce Lindsay & Runze Li, 2012. "Local modal regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(3), pages 647-663.
    4. Yatchew, A., 1997. "An elementary estimator of the partial linear model," Economics Letters, Elsevier, vol. 57(2), pages 135-143, December.
    5. Weixin Yao & Longhai Li, 2014. "A New Regression Model: Modal Linear Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(3), pages 656-671, September.
    6. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, January.
    7. Riquan Zhang & Weihua Zhao & Jicai Liu, 2013. "Robust estimation and variable selection for semiparametric partially linear varying coefficient model based on modal regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(2), pages 523-544, June.
    8. 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.
    9. Song, Weixing & Yao, Weixin & Xing, Yanru, 2014. "Robust mixture regression model fitting by Laplace distribution," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 128-137.
    10. Runze Li & Lei Nie, 2008. "Efficient Statistical Inference Procedures for Partially Nonlinear Models and their Applications," Biometrics, The International Biometric Society, vol. 64(3), pages 904-911, September.
    11. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, January.
    12. 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.
    13. Song, Lixin & Zhao, Yue & Wang, Xiaoguang, 2010. "Sieve least squares estimation for partially nonlinear models," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1271-1283, September.
    14. 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.
    15. Inyoung Kim & Noah D. Cohen & Raymond J. Carroll, 2003. "Semiparametric Regression Splines in Matched Case-Control Studies," Biometrics, The International Biometric Society, vol. 59(4), pages 1158-1169, December.
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    Cited by:

    1. Rui Li & Yuanyuan Zhang, 2021. "Two-stage estimation and simultaneous confidence band in partially nonlinear additive model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1109-1140, November.

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