Exact Quadratic Error of the Local Linear Regression Operator Estimator for Functional Covariates

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

Abstract

In this paper, it is studied the asymptotic behavior of the nonparametric local linear estimation of the regression operator when the covariates are curves. Under some general conditions we give the exact expression involved in the leading terms of the quadratic error of this estimator. The obtained results affirm the superiority of the local linear modeling over the kernel method, in functional statistics framework.

References

  1. 1.
    Barrientos-Marin, J., Ferraty, F., Vieu, P.: Locally modelled regression and functional data. J. Nonparametr. Stat. 22, 617–632 (2010)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Baìllo, A., Grané, A.: Local linear regression for functional predictor and scalar response. J. Multivar. Anal. 100, 102–111 (2009)MATHCrossRefGoogle Scholar
  3. 3.
    Chu, C.-K., Marron, J.-S.: Choosing a kernel regression estimator. Stat. Sci. 6, 404–436 (1991)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Davidian, M., Lin, X., Wang, J.L.: Introduction. Stat. Sinica 14, 613–614 (2004)MathSciNetGoogle Scholar
  5. 5.
    Demongeot, J., Laksaci, A., Madani, F., Rachdi, M.: Functional data: local linear estimation of the conditional density and its application. Statistics 47, 26–44 (2013)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    El Methni, M., Rachdi, M.: Local weighted average estimation of the regression operator for functional data. Commun. Stat. Theory Methods 40, 3141–3153 (2010)CrossRefGoogle Scholar
  7. 7.
    Fan, J.: Design-adaptive nonparametric regression. J. Am. Stat. Assoc. 87, 998–1004 (1992)MATHCrossRefGoogle Scholar
  8. 8.
    Fan, J., Gijbels, I.: Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability, vol. 66. Chapman & Hall, London (1996)Google Scholar
  9. 9.
    Fan, J., Yao, Q.: Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York (2003)CrossRefGoogle Scholar
  10. 10.
    Ferraty, F.: High-dimensional data: a fascinating statistical challenge. J. Multivar. Anal. 101, 305–30 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ferraty, F., Romain, Y.: The Oxford Handbook of Functional Data Analysis. Oxford University Press, Oxford (2011)Google Scholar
  12. 12.
    Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics, Springer, New York (2006)Google Scholar
  13. 13.
    Ferraty, F., Mas, A., Vieu, P.: Nonparametric regression on functional data: inference and practical aspects. Aust. N. Z. J. Stat. 49, 267–286 (2007)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferraty, F., Laksaci, A., Tadj, A., Vieu, P.: Rate of uniform consistency for nonparametric estimates with functional variables. J. Stat. Plan. Inference 140, 335–352 (2010)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Gonzalez Manteiga, W., Vieu, P.: Statistics for functional data. Comput. Stat. Data Anal. 51, 4788–4792 (2007MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Sarda, P., Vieu, P.: Kernel regression. In: Schimek, M.G. (ed.) Smoothing and Regression: Approaches, Computation and Application. Wiley Series in Probability and Statistics, pp. 43–70. Wiley, Chichester, New York (2000)Google Scholar
  17. 17.
    Valderrama, M.J.: An overview to modelling functional data. Comput. Stat. 22, 331–334 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Laboratoire de Statistique et Processus StochastiquesUniversité Djillali LiabèsSidi Bel-AbbèsAlgeria
  2. 2.Laboratoire AGIM FRE 3405 CNRSUniversity of Grenoble-AlpesSaint-Martin-d’HèresFrance
  3. 3.University of P. Mendès France (Grenoble 2), UFR SHSGrenoble Cedex 09France

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