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Smoothing splines with varying smoothing parameter

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  • Xiao Wang
  • Pang Du
  • Jinglai Shen

Abstract

This paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain. Two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty. The penalty is a function of the design points in order to accommodate local behaviour of the regression function. We show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator, and that the equivalent kernel is spatially dependent. The equivalent kernels for traditional smoothing splines are a special case of this general solution. With the aid of the Green's function for a two-point boundary value problem, explicit forms of the asymptotic mean and variance are obtained for any interior point. Thus, the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error. Simulation results and an application illustrate the performance of the proposed estimator. Copyright 2013, Oxford University Press.

Suggested Citation

  • Xiao Wang & Pang Du & Jinglai Shen, 2013. "Smoothing splines with varying smoothing parameter," Biometrika, Biometrika Trust, vol. 100(4), pages 955-970.
    • Handle: RePEc:oup:biomet:v:100:y:2013:i:4:p:955-970
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    File URL: http://hdl.handle.net/10.1093/biomet/ast031
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    Cited by:

    1. Martina Čampulová, 2018. "Comparison of Methods for Smoothing Environmental Data with an Application to Particulate Matter PM10," Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, Mendel University Press, vol. 66(2), pages 453-463.
    2. Yang, Lianqiang & Hong, Yongmiao, 2017. "Adaptive penalized splines for data smoothing," Computational Statistics & Data Analysis, Elsevier, vol. 108(C), pages 70-83.
    3. Tong, Xiaojun & He, Zhuoqiong Chong & Sun, Dongchu, 2018. "Estimating Chinese Treasury yield curves with Bayesian smoothing splines," Econometrics and Statistics, Elsevier, vol. 8(C), pages 94-124.
    4. Soumya D. Mohanty & Ethan Fahnestock, 2021. "Adaptive spline fitting with particle swarm optimization," Computational Statistics, Springer, vol. 36(1), pages 155-191, March.
    5. Fabio Centofanti & Antonio Lepore & Alessandra Menafoglio & Biagio Palumbo & Simone Vantini, 2023. "Adaptive smoothing spline estimator for the function-on-function linear regression model," Computational Statistics, Springer, vol. 38(1), pages 191-216, March.
    6. Kim, Daeju & Kawano, Shuichi & Ninomiya, Yoshiyuki, 2023. "Smoothly varying regularization," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).

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