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CLT in functional linear regression models
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  • Published: 23 August 2006

CLT in functional linear regression models

  • Hervé Cardot1,
  • André Mas2 &
  • Pascal Sarda3,4 

Probability Theory and Related Fields volume 138, pages 325–361 (2007)Cite this article

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A Correction to this article was published on 23 June 2023

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Abstract

We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made.

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  • 23 June 2023

    A Correction to this paper has been published: https://doi.org/10.1007/s00440-023-01215-7

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Author information

Authors and Affiliations

  1. CESAER, 26, bd Docteur Petitjean, BP 87999, 21079, Dijon Cedex, France

    Hervé Cardot

  2. Institut de Modélisation Mathématique de Montpellier, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier Cedex 5, France

    André Mas

  3. Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse Cedex 4, France

    Pascal Sarda

  4. GRIMM, EA 2254, Université Toulouse-le-Mirail, 5, Allées Antonio-Machado, 31058, Toulouse Cedex 1, France

    Pascal Sarda

Authors
  1. Hervé Cardot
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  2. André Mas
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  3. Pascal Sarda
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Corresponding author

Correspondence to André Mas.

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Cite this article

Cardot, H., Mas, A. & Sarda, P. CLT in functional linear regression models. Probab. Theory Relat. Fields 138, 325–361 (2007). https://doi.org/10.1007/s00440-006-0025-2

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  • Received: 12 June 2005

  • Revised: 07 July 2006

  • Published: 23 August 2006

  • Issue Date: July 2007

  • DOI: https://doi.org/10.1007/s00440-006-0025-2

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Keywords

  • Central limit theorem
  • Hilbertian random variables
  • Functional data analysis
  • Covariance operator
  • Inverse problem
  • Regularization
  • Perturbation theory

Mathematics Subject Classification (2000)

  • 60F05
  • 60H05
  • 45Q05
  • 62J07
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