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On rates of convergence in functional linear regression

Author

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  • Li, Yehua
  • Hsing, Tailen

Abstract

This paper investigates the rate of convergence of estimating the regression weight function in a functional linear regression model. It is assumed that the predictor as well as the weight function are smooth and periodic in the sense that the derivatives are equal at the boundary points. Assuming that the functional data are observed at discrete points with measurement error, the complex Fourier basis is adopted in estimating the true data and the regression weight function based on the penalized least-squares criterion. The rate of convergence is then derived for both estimators. A simulation study is also provided to illustrate the numerical performance of our approach, and to make a comparison with the principal component regression approach.

Suggested Citation

  • Li, Yehua & Hsing, Tailen, 2007. "On rates of convergence in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1782-1804, October.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:9:p:1782-1804
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    References listed on IDEAS

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    1. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167.
    2. Cardot, Hervé & Sarda, Pacal, 2005. "Estimation in generalized linear models for functional data via penalized likelihood," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 24-41, January.
    3. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506.
    4. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
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    Citations

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    Cited by:

    1. Peter Hall & Giles Hooker, 2016. "Truncated linear models for functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(3), pages 637-653, June.
    2. repec:eee:jmvana:v:163:y:2018:i:c:p:15-36 is not listed on IDEAS
    3. Siegfried Hörmann & Łukasz Kidziński, 2015. "A Note on Estimation in Hilbertian Linear Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 43-62, March.
    4. repec:eee:jmvana:v:161:y:2017:i:c:p:68-82 is not listed on IDEAS
    5. Huang, Lele & Zhao, Junlong & Wang, Huiwen & Wang, Siyang, 2016. "Robust shrinkage estimation and selection for functional multiple linear model through LAD loss," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 384-400.
    6. Shin, Hyejin & Lee, Myung Hee, 2012. "On prediction rate in partial functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 93-106, January.
    7. Apanasovich, Tatiyana V. & Goldstein, Edward, 2008. "On prediction error in functional linear regression," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1807-1810, September.
    8. Chiou, Jeng-Min & Yang, Ya-Fang & Chen, Yu-Ting, 2016. "Multivariate functional linear regression and prediction," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 301-312.
    9. Brunel, Élodie & Mas, André & Roche, Angelina, 2016. "Non-asymptotic adaptive prediction in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 208-232.
    10. Zhou, Jianjun & Chen, Min, 2012. "Spline estimators for semi-functional linear model," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 505-513.
    11. Shujie Ma, 2016. "Estimation and inference in functional single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 181-208, February.
    12. Fraiman, Ricardo & Gimenez, Yanina & Svarc, Marcela, 2016. "Feature selection for functional data," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 191-208.
    13. Jianjun Zhou & Zhao Chen & Qingyan Peng, 2016. "Polynomial spline estimation for partial functional linear regression models," Computational Statistics, Springer, vol. 31(3), pages 1107-1129, September.
    14. Shujie Ma, 2016. "Estimation and inference in functional single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 181-208, February.
    15. repec:bla:istatr:v:85:y:2017:i:1:p:61-83 is not listed on IDEAS

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