On the Effect of Noisy Observations of the Regressor in a Functional Linear Model

Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


We consider the estimation of the slope function in functional linear regression, where a scalar response Y is modeled in dependence of a random function X, when Yand only a panel Z 1Z L of noisy observations of X are observable. Assuming an iid. sample of (Y,Z 1Z L) we derive in terms of both, the sample size and the panel size, a lower bound of a maximal weigthed risk over certain ellipsoids of slope functions.We prove that a thresholded projection estimator can attain the lower bound up to a constant.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bereswill, M., Johannes, J.: On the effect of noisy observations of the regressor in a functional linear model. Technical report, Universit´e catholique de Louvain (2010) 2. Bosq. D.: Linear Processes in Function Spaces. Lecture Notes in Statistics, 149, Springer- Verlag (2000)Google Scholar
  2. 2.
    Cardot, H., Johannes, J.: Thresholding projection estimators in functional linear models. J. Multivariate Anal. 101 (2), 395–408 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cardot, H., Ferraty, F., Sarda, P.: Functional linear model. Stat. Probabil. Lett. 45, 11–22 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cardot, H., Ferraty, F., Sarda, P.: Spline estimators for the functional linear model. Stat. Sinica 13 571–591 (2003)MathSciNetMATHGoogle Scholar
  5. 5.
    Cardot, H., Ferraty, F., Mas, A., Sarda, P.: Clt in functional linear regression models. Probab. Theor. Rel. 138, 325–361 (2007)MATHCrossRefGoogle Scholar
  6. 6.
    Crambes, C., Kneip, A., Sarda, P.: Smoothing splines estimators for functional linear regression. Ann. Stat. 37 (1), 35–72 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Efromovich, S., Koltchinskii, V.: On inverse problems with unknown operators. IEEE T. Inform. Theory 47 (7), 2876–2894 (2001)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Practice and Theory. Springer, New York (2006)MATHGoogle Scholar
  9. 9.
    Hall, P., Horowitz, J. L.: Methodology and convergence rates for functional linear regression. Ann. Stat. 35 (1), 70–91 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hoffmann, M., Reiß, M.: Nonlinear estimation for linear inverse problems with error in the operator. Ann. Stat. 36 (1), 310–336 (2008)MATHCrossRefGoogle Scholar
  11. 11.
    Marx, B. D., Eilers, P. H.: Generalized linear regression on sampled signals and curves: a p-spline approach. Technometrics 41, 1–13 (1999)CrossRefGoogle Scholar
  12. 12.
    M¨uller, H.-G., Stadtm¨uller, U.: Generalized functional linear models. Ann. Stat. 33 (2), 774– 805 (2005)Google Scholar
  13. 13.
    Natterer, F.: Error bounds for Tikhonov regularization in Hilbert scales. Appl. Anal. 18, 29–37 (1984)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ramsay, J., Silverman, B. Functional Data Analysis (Second Edition). Springer, New York (2005)Google Scholar
  15. 15.
    Yao, F.,M¨uller, H.-G.,Wang, J.-L.: Functional linear regression analysis for longitudinal data. Ann. Stat. 33 (6), 2873–2903 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.University of HeidelbergHeidelbergGermany
  2. 2.Université Catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations