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2013 On rate optimal local estimation in functional linear regression
Jan Johannes, Rudolf Schenk
Electron. J. Statist. 7: 191-216 (2013). DOI: 10.1214/13-EJS767


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.


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Jan Johannes. Rudolf Schenk. "On rate optimal local estimation in functional linear regression." Electron. J. Statist. 7 191 - 216, 2013.


Published: 2013
First available in Project Euclid: 24 January 2013

zbMATH: 1337.62161
MathSciNet: MR3020418
Digital Object Identifier: 10.1214/13-EJS767

Primary: 62J05
Secondary: 62G05, 62J20

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society


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