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Local smoothing regression with functional data

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Abstract

Kernel estimates of a regression operator are investigated when the explanatory variable is of functional type. The bandwidths are locally chosen by a data-driven method based on the minimization of a functional version of a cross-validated criterion. A short asymptotic theoretical support is provided and the main body of this paper is devoted to various finite sample size applications. In particular, it is shown through some simulations, that a local bandwidth choice enables to capture some underlying heterogeneous structures in the functional dataset. As a consequence, the estimation of the relationship between a functional variable and a scalar response, and hence the prediction, can be significantly improved by using local smoothing parameter selection rather than global one. This is also confirmed from a chemometrical real functional dataset. These improvements are much more important than in standard finite dimensional setting.

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References

  • Cardot H, Ferraty F, Sarda P (2003) Spline estimators for the functional linear model. Stat Sin 13(3):571–591

    MATH  MathSciNet  Google Scholar 

  • Ferraty F, Vieu P (2004) Nonparametric models for functional data, with application in regression, time series prediction and curve discrimination. J Nonparametric Stat 16(1–2):111–125

    Article  MATH  MathSciNet  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, Springer Series in Statistics, New York

    MATH  Google Scholar 

  • Grenander U (1963) Probabilities on algebraic structures. Wiley, New York

    MATH  Google Scholar 

  • Härdle W, Bowman A (1988) Bootrapping in nonparametric regression: local adaptive smoothing and confidence bands. J Am Stat Assoc 83:102–110

    Article  MATH  Google Scholar 

  • Härdle W, Marron JS (1985) Optimal bandwidth selection in nonparametric regression function estimation. Ann Stat 13(4):1465–1481

    MATH  Google Scholar 

  • Marron JS, Härdle W (1986) Random approximations to some measures of accuracy in nonparametric curve estimation. J Multivar Anal 20:91–113

    Article  MATH  Google Scholar 

  • Müller HG, Stadtmüller U (1987) Variable bandwidth kernel estimators of regression curves. Ann Stat 15:182–201

    MATH  Google Scholar 

  • Nadaraya EA (1964) On estimating regression. Theory Probab. Appl. 9:141–142

    Article  Google Scholar 

  • Rachdi M, Vieu P (2006) Nonparametric regression for functional data: automatic smoothing parameter selection. J Stat Plann Inference (in press)

  • Ramsay J, Silverman B (1997) Functional data analysis. Springer, Springer Series in Statistics, New York

    MATH  Google Scholar 

  • Ramsay J, Silverman B (2002) Applied functional data analysis. Methods and case studies. Springer, Springer Series in Statistics, New York

    MATH  Google Scholar 

  • Vieu P (1991) Nonparametric regression: optimal local bandwidth choice. J R Stat Soc 53(2): 453–464

    MATH  MathSciNet  Google Scholar 

  • Watson GS (1964) Smooth regression analysis. Sankhya Ser. A 26:359–372

    MATH  MathSciNet  Google Scholar 

  • Whittle P (1960) Bounds for the moments of linear and quadratic forms in independent random variables (in russian). Teor. Verojatnost. I Primenen, 331–335. Engl. Transl. Theory Probab. Appl. 5, pp 302–305

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Authors and Affiliations

  1. Université de Grenoble, LJK UMR CNRS 5224, UFR SHS, BP. 47, F38040, Grenoble Cedex 09, France

    K. Benhenni & M. Rachdi

  2. Université Paul Sabatier, LSP UMR CNRS 5583, 118, Route de Narbonne, 31062, Toulouse Cedex, France

    F. Ferraty & P. Vieu

Authors
  1. K. Benhenni
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  2. F. Ferraty
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  3. M. Rachdi
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  4. P. Vieu
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Correspondence to M. Rachdi.

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Benhenni, K., Ferraty, F., Rachdi, M. et al. Local smoothing regression with functional data. Computational Statistics 22, 353–369 (2007). https://doi.org/10.1007/s00180-007-0045-0

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  • DOI: https://doi.org/10.1007/s00180-007-0045-0

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