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Functional linear model

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  • Cardot, Hervé
  • Ferraty, Frédéric
  • Sarda, Pascal
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    Abstract

    In this paper, we study a regression model in which explanatory variables are sampling points of a continuous-time process. We propose an estimator of regression by means of a Functional Principal Component Analysis analogous to the one introduced by Bosq [(1991) NATO, ASI Series, pp. 509-529] in the case of Hilbertian AR processes. Both convergence in probability and almost sure convergence of this estimator are stated.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 45 (1999)
    Issue (Month): 1 (October)
    Pages: 11-22

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    Handle: RePEc:eee:stapro:v:45:y:1999:i:1:p:11-22

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    Related research

    Keywords: Functional linear model Functional data analysis Hilbert spaces Convergence;

    References

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    1. 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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    Citations

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    Cited by:
    1. F. Ferraty & A. Goia & E. Salinelli & P. Vieu, 2013. "Functional projection pursuit regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 22(2), pages 293-320, June.
    2. Cardot, Hervé & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 395-408, February.
    3. Dalia Jazmín Valencia García & Rosa E. Lillo & Juan Romo, 2013. "A Kendall correlation coefficient for functional dependence," Statistics and Econometrics Working Papers ws133228, Universidad Carlos III, Departamento de Estadística y Econometría.
    4. Guillas, Serge, 2001. "Rates of convergence of autocorrelation estimates for autoregressive Hilbertian processes," Statistics & Probability Letters, Elsevier, vol. 55(3), pages 281-291, December.
    5. Nerini, David & Ghattas, Badih, 2007. "Classifying densities using functional regression trees: Applications in oceanology," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4984-4993, June.
    6. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2004. "An anova test for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 111-122, August.
    7. Cardot, Hervé & Sarda, Pacal, 2005. "Estimation in generalized linear models for functional data via penalized likelihood," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 24-41, January.
    8. Lee, Eun Ryung & Park, Byeong U., 2012. "Sparse estimation in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 1-17.
    9. Cardot, Herve & Crambes, Christophe & Kneip, Alois & Sarda, Pascal, 2007. "Smoothing splines estimators in functional linear regression with errors-in-variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4832-4848, June.
    10. Shin, Hyejin & Hsing, Tailen, 2012. "Linear prediction in functional data analysis," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3680-3700.
    11. Febrero-Bande, Manuel & Galeano, Pedro & González-Manteiga, Wenceslao, 2010. "Measures of influence for the functional linear model with scalar response," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 327-339, February.
    12. Ferraty, F. & Vieu, P., 2003. "Curves discrimination: a nonparametric functional approach," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 161-173, October.
    13. Han Lin Shang, 2011. "A survey of functional principal component analysis," Monash Econometrics and Business Statistics Working Papers 6/11, Monash University, Department of Econometrics and Business Statistics.
    14. Nerini, David & Monestiez, Pascal & Manté, Claude, 2010. "Cokriging for spatial functional data," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 409-418, February.
    15. Mareike Bereswill & Jan Johannes, 2013. "On the effect of noisy measurements of the regressor in functional linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 22(3), pages 488-513, September.
    16. Yao, Fang, 2007. "Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 40-56, January.
    17. Hlubinka, Daniel & Prchal, Lubos, 2007. "Changes in atmospheric radiation from the statistical point of view," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4926-4941, June.
    18. Cristian Preda & Gilbert Saporta & Caroline Lévéder, 2007. "PLS classification of functional data," Computational Statistics, Springer, vol. 22(2), pages 223-235, July.
    19. Goia, Aldo & May, Caterina & Fusai, Gianluca, 2010. "Functional clustering and linear regression for peak load forecasting," International Journal of Forecasting, Elsevier, vol. 26(4), pages 700-711, October.
    20. Herve Cardot & Robert Faivre & Michel Goulard, 2003. "Functional approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(10), pages 1185-1199.
    21. Zhou, Jianjun & Chen, Min, 2012. "Spline estimators for semi-functional linear model," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 505-513.
    22. Amato, U. & Antoniadis, A. & De Feis, I., 2006. "Dimension reduction in functional regression with applications," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2422-2446, May.
    23. Qi, Xin & Zhao, Hongyu, 2011. "Some theoretical properties of Silverman's method for Smoothed functional principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 741-767, April.

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