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Thresholding projection estimators in functional linear models

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  • Cardot, Hervé
  • Johannes, Jan

Abstract

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows us to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits us to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove that these estimators are minimax and rates of convergence are given for some particular cases.

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  • Cardot, Hervé & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 395-408, February.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:2:p:395-408
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    1. Cardot, Hervé & Ferraty, Frédéric & Sarda, Pascal, 1999. "Functional linear model," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 11-22, October.
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    7. Hardle, Wolfgang & Tsybakov, A. B., 1993. "How sensitive are average derivatives?," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 31-48, July.
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    Cited by:

    1. Jean-Pierre FLORENS & Joel L. HOROWITZ & Ingrid VAN KEILEGOM, 2017. "Bias-Corrected Confidence Intervals in a Class of Linear Inverse Problems," Annals of Economics and Statistics, GENES, issue 128, pages 203-228.
    2. Centorrino Samuele & Feve Frederique & Florens Jean-Pierre, 2017. "Additive Nonparametric Instrumental Regressions: A Guide to Implementation," Journal of Econometric Methods, De Gruyter, vol. 6(1), pages 1-25, January.
    3. Imaizumi, Masaaki & Kato, Kengo, 2018. "PCA-based estimation for functional linear regression with functional responses," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 15-36.
    4. Bereswill, Mareike & Johannes, Jan, 2011. "On the effect of noisy observations of the regressor in a functional linear model," LIDAM Discussion Papers ISBA 2011039, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Mareike Bereswill & Jan Johannes, 2013. "On the effect of noisy measurements of the regressor in functional linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 488-513, September.
    6. 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.
    7. Shin, Hyejin & Hsing, Tailen, 2012. "Linear prediction in functional data analysis," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3680-3700.
    8. 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.
    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. Lee, Eun Ryung & Park, Byeong U., 2012. "Sparse estimation in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 1-17.
    11. Florens, Jean-Pierre & Van Bellegem, Sébastien, 2015. "Instrumental variable estimation in functional linear models," Journal of Econometrics, Elsevier, vol. 186(2), pages 465-476.
    12. Florens, Jean-Pierre & Horowitz, Joel & Van Keilegom, Ingrid, 2016. "Bias-corrected condence intervals in a class of linear inverse problems," LIDAM Discussion Papers ISBA 2016021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. Jean-Pierre Florens & Joel L. Horowitz & Ingred van Keilegom, 2016. "Bias-corrected confidence intervals in a class of linear inverse problems," CeMMAP working papers 19/16, Institute for Fiscal Studies.
    14. Van Bellegem, Sébastien & Florens, Jean-Pierre, 2014. "Instrumental variable estimation in functional linear models," LIDAM Discussion Papers CORE 2014056, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. Manuel Febrero-Bande & Pedro Galeano & Wenceslao González-Manteiga, 2017. "Functional Principal Component Regression and Functional Partial Least-squares Regression: An Overview and a Comparative Study," International Statistical Review, International Statistical Institute, vol. 85(1), pages 61-83, April.

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