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On the effect of noisy measurements of the regressor in functional linear models

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  • Mareike Bereswill
  • Jan Johannes

Abstract

We consider the estimation of the slope function in functional linear regression, where a scalar response Y is modelled in dependence of a random function X, when Y and only a panel Z 1 ,…,Z L of noisy measurements of X are observable. Assuming an i.i.d. sample of (Y,Z 1 ,…,Z L ) of size n we propose an estimator of the slope which is based on a dimension reduction technique and additional thresholding. We derive in terms of both the sample size n and the panel size L a lower bound of a maximal weighted risk over a certain ellipsoid of slope functions and a certain class of covariance operators associated with the regressor X. It is shown that the proposed estimator attains this lower bound up to a constant and hence it is minimax-optimal. The results are illustrated considering different configurations which cover in particular the estimation of the slope as well as its derivatives. Copyright Sociedad de Estadística e Investigación Operativa 2013

Suggested Citation

  • 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;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 488-513, September.
    • Handle: RePEc:spr:testjl:v:22:y:2013:i:3:p:488-513
    DOI: 10.1007/s11749-013-0325-7
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    References listed on IDEAS

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    1. Cardot, Hervé & Ferraty, Frédéric & Sarda, Pascal, 1999. "Functional linear model," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 11-22, October.
    2. Preda, C. & Saporta, G., 2005. "Clusterwise PLS regression on a stochastic process," Computational Statistics & Data Analysis, Elsevier, vol. 49(1), pages 99-108, April.
    3. Cardot, Herve & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," LIDAM Reprints ISBA 2010034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Mario Forni & Lucrezia Reichlin, 1998. "Let's Get Real: A Factor Analytical Approach to Disaggregated Business Cycle Dynamics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 453-473.
    5. Cardot, Hervé & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 395-408, February.
    6. Johannes, Jan & Schenk, Rudolf, 2013. "On rate optimal local estimation in functional linear regression," LIDAM Reprints ISBA 2013014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Preda, C. & Saporta, G., 2005. "PLS regression on a stochastic process," Computational Statistics & Data Analysis, Elsevier, vol. 48(1), pages 149-158, January.
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