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Regression when both response and predictor are functions

Author

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  • Ferraty, F.
  • Van Keilegom, I.
  • Vieu, P.

Abstract

We consider a nonparametric regression model where the response Y and the covariate X are both functional (i.e. valued in some infinite-dimensional space). We define a kernel type estimator of the regression operator and we first establish its pointwise asymptotic normality. The double functional feature of the problem makes the formulas of the asymptotic bias and variance even harder to estimate than in more standard regression settings, and we propose to overcome this difficulty by using resampling ideas. Both a naive and a wild componentwise bootstrap procedure are studied, and their asymptotic validity is proved. These results are also extended to data-driven bases which is a key point for implementing this methodology. The theoretical advances are completed by some simulation studies showing both the practical feasibility of the method and the good behavior for finite sample sizes of the kernel estimator and of the bootstrap procedures to build functional pseudo-confidence area.

Suggested Citation

  • Ferraty, F. & Van Keilegom, I. & Vieu, P., 2012. "Regression when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 10-28.
    • Handle: RePEc:eee:jmvana:v:109:y:2012:i:c:p:10-28
    DOI: 10.1016/j.jmva.2012.02.008
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    References listed on IDEAS

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    1. Masry, Elias, 2005. "Nonparametric regression estimation for dependent functional data: asymptotic normality," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 155-177, January.
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    4. Jaromir Antoch & Lubos Prchal & Maria Rosaria De Rosa & Pascal Sarda, 2010. "Electricity consumption prediction with functional linear regression using spline estimators," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(12), pages 2027-2041.
    5. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
    6. Yurinskii, V. V., 1976. "Exponential inequalities for sums of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 6(4), pages 473-499, December.
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