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A Note on Estimation in Hilbertian Linear Models

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  • Siegfried Hörmann
  • Łukasz Kidziński

Abstract

type="main" xml:id="sjos12094-abs-0001"> We study estimation and prediction in linear models where the response and the regressor variable both take values in some Hilbert space. Our main objective is to obtain consistency of a principal component-based estimator for the regression operator under minimal assumptions. In particular, we avoid some inconvenient technical restrictions that have been used throughout the literature. We develop our theory in a time-dependent setup that comprises as important special case the autoregressive Hilbertian model.

Suggested Citation

  • Siegfried Hörmann & Łukasz Kidziński, 2015. "A Note on Estimation in Hilbertian Linear Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 43-62, March.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:1:p:43-62
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    File URL: http://hdl.handle.net/10.1111/sjos.12094
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    References listed on IDEAS

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    1. Cardot, Hervé & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 395-408, February.
    2. Cardot, Hervé & Ferraty, Frédéric & Sarda, Pascal, 1999. "Functional linear model," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 11-22, October.
    3. Li, Yehua & Hsing, Tailen, 2007. "On rates of convergence in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1782-1804, October.
    4. Reiss, Philip T. & Ogden, R. Todd, 2007. "Functional Principal Component Regression and Functional Partial Least Squares," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 984-996, September.
    5. 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.
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    Cited by:

    1. Petrovich, Justin & Reimherr, Matthew, 2017. "Asymptotic properties of principal component projections with repeated eigenvalues," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 42-48.

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