On the effect of noisy measurements of the regressor in functional linear models
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Mareike Bereswill ; Jan Johannes [1]
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[1] Université Catholique de Louvain
Université Catholique de Louvain
Arrondissement de Nivelles, Bélgica
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- Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 22, Nº. 3, 2013, págs. 488-513
- Idioma: inglés
- DOI: 10.1007/s11749-013-0325-7
- Texto completo no disponible (Saber más ...)
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Resumen
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.
