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Theory for penalised spline regression

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  • Peter Hall
  • J. D. Opsomer

Abstract

Penalised spline regression is a popular new approach to smoothing, but its theoretical properties are not yet well understood. In this paper, mean squared error expressions and consistency results are derived by using a white-noise model representation for the estimator. The effect of the penalty on the bias and variance of the estimator is discussed, both for general splines and for the case of polynomial splines. The penalised spline regression estimator is shown to achieve the optimal nonparametric convergence rateestablished by Stone (1982). Copyright 2005, Oxford University Press.

Suggested Citation

  • Peter Hall & J. D. Opsomer, 2005. "Theory for penalised spline regression," Biometrika, Biometrika Trust, vol. 92(1), pages 105-118, March.
  • Handle: RePEc:oup:biomet:v:92:y:2005:i:1:p:105-118
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    File URL: http://hdl.handle.net/10.1093/biomet/92.1.105
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    References listed on IDEAS

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    1. Cressie, Noel & Davidson, Jennifer L., 1998. "Image analysis with partially ordered markov models," Computational Statistics & Data Analysis, Elsevier, vol. 29(1), pages 1-26, November.
    2. Ming Gao Gu & Hong-Tu Zhu, 2001. "Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 339-355.
    3. Francesco Bartolucci, 2002. "A recursive algorithm for Markov random fields," Biometrika, Biometrika Trust, vol. 89(3), pages 724-730, August.
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    Cited by:

    1. Wahba, Jackline & Schluter, Christian, 2009. "Illegal migration, wages and remittances- semi-parametric estimation of illegality effects," Discussion Paper Series In Economics And Econometrics 0913, Economics Division, School of Social Sciences, University of Southampton.
    2. Takuma Yoshida, 2016. "Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(4), pages 278-303, November.
    3. Chen, Haiqiang & Fang, Ying & Li, Yingxing, 2015. "Estimation And Inference For Varying-Coefficient Models With Nonstationary Regressors Using Penalized Splines," Econometric Theory, Cambridge University Press, vol. 31(04), pages 753-777, August.
    4. Wahba, Jackline & Schluter, Christian, 2009. "Illegal migration, wages and remittances- semi-parametric estimation of illegality effects," Discussion Paper Series In Economics And Econometrics 913, Economics Division, School of Social Sciences, University of Southampton.
    5. Simon N. Wood, 2011. "Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 3-36, January.
    6. Holland, Ashley D., 2017. "Penalized spline estimation in the partially linear model," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 211-235.
    7. Schluter, Christian & Wahba, Jackline, 2009. "Illegal Migration, Wages, and Remittances: Semi-Parametric Estimation of Illegality Effects," IZA Discussion Papers 4527, Institute for the Study of Labor (IZA).
    8. Lee, Wang-Sheng, 2014. "Big and Tall: Is there a Height Premium or Obesity Penalty in the Labor Market?," IZA Discussion Papers 8606, Institute for the Study of Labor (IZA).
    9. Wu, Ximing & Sickles, Robin, 2014. "Semiparametric Estimation under Shape Constraints," Working Papers 15-021, Rice University, Department of Economics.
    10. repec:wyi:journl:002195 is not listed on IDEAS
    11. Christian Schluter & Jackline Wahba, 2012. "Abstract: Illegal Migration, Wages, and Remittances: Semi-Parametric Estimation of Illegality Effects," Norface Discussion Paper Series 2012037, Norface Research Programme on Migration, Department of Economics, University College London.
    12. Sima, Diana M. & Van Huffel, Sabine, 2006. "A class of template splines," Computational Statistics & Data Analysis, Elsevier, vol. 50(12), pages 3486-3499, August.
    13. Lee, Wang-Sheng, 2014. "Is the BMI a Relic of the Past?," IZA Discussion Papers 8637, Institute for the Study of Labor (IZA).
    14. Luo Xiao & Yingxing Li & David Ruppert, 2013. "Fast bivariate P-splines: the sandwich smoother," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 577-599, June.
    15. Göran Kauermann & Tatyana Krivobokova & Ludwig Fahrmeir, 2009. "Some asymptotic results on generalized penalized spline smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 487-503.

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