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M-estimation for the partially linear regression model under monotonic constraints


  • Du, Jiang
  • Sun, Zhimeng
  • Xie, Tianfa


In this paper, we study M-estimation for the partially linear model under monotonic constraints. We use monotone B-splines to approximate the monotone nonparametric function. We show the large sample properties of the resulting estimators. The proposed estimator of parameter part is root-n consistent, and asymptotically normal and the estimator for the nonparametric component achieves the optimal convergence rate. A simulation study is conducted to evaluate the finite sample performance of the method. The proposed procedure is illustrated by an air pollution study.

Suggested Citation

  • Du, Jiang & Sun, Zhimeng & Xie, Tianfa, 2013. "M-estimation for the partially linear regression model under monotonic constraints," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1353-1363.
  • Handle: RePEc:eee:stapro:v:83:y:2013:i:5:p:1353-1363
    DOI: 10.1016/j.spl.2013.01.006

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    References listed on IDEAS

    1. Minggen Lu & Ying Zhang & Jian Huang, 2007. "Estimation of the mean function with panel count data using monotone polynomial splines," Biometrika, Biometrika Trust, vol. 94(3), pages 705-718.
    2. Sun, Zhimeng & Zhang, Zhongzhan, 2013. "Semiparametric analysis of additive isotonic errors-in-variables regression models," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 100-114.
    3. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, April.
    4. He, Xuming & Fung, Wing K. & Zhu, Zhongyi, 2005. "Robust Estimation in Generalized Partial Linear Models for Clustered Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1176-1184, December.
    5. Carroll, Raymond J. & Delaigle, Aurore & Hall, Peter, 2011. "Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 191-202.
    6. Arcones, Miguel A., 1996. "The Bahadur-Kiefer Representation of Lp Regression Estimators," Econometric Theory, Cambridge University Press, vol. 12(02), pages 257-283, June.
    7. Zeckhauser, Richard & Thompson, Mark, 1970. "Linear Regression with Non-Normal Error Terms," The Review of Economics and Statistics, MIT Press, vol. 52(3), pages 280-286, August.
    8. He, Xuming & Shao, Qi-Man, 2000. "On Parameters of Increasing Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 120-135, April.
    9. Xuming He, 2002. "Estimation in a semiparametric model for longitudinal data with unspecified dependence structure," Biometrika, Biometrika Trust, vol. 89(3), pages 579-590, August.
    10. Lu, Minggen, 2010. "Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2528-2542, November.
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