Electronic Journal of Statistics
- Electron. J. Statist.
- Volume 7 (2013), 191-216.
On rate optimal local estimation in functional linear regression
We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of random functions. The theory in this paper covers in particular point-wise estimation as well as the estimation of weighted averages of the slope parameter. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent under mild assumptions. We derive a lower bound for the maximal mean squared error of any estimator over a certain ellipsoid of slope parameters and a certain class of covariance operators associated with the regressor. It is shown that the proposed estimator attains this lower bound up to a constant and hence it is minimax optimal. Our results are appropriate to discuss a wide range of possible regressors, slope parameters and functionals. They are illustrated by considering the point-wise estimation of the slope parameter or its derivatives and its average value over a given interval.
Electron. J. Statist., Volume 7 (2013), 191-216.
First available in Project Euclid: 24 January 2013
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Johannes, Jan; Schenk, Rudolf. On rate optimal local estimation in functional linear regression. Electron. J. Statist. 7 (2013), 191--216. doi:10.1214/13-EJS767. https://projecteuclid.org/euclid.ejs/1359041589