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

Mareike Bereswill () and Jan Johannes ()

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2013, vol. 22, issue 3, 488-513

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

Keywords: Minimax-optimal estimation; Linear Galerkin approach; Linear inverse problem; Sobolev space; 62J05; 62G20; 62G08 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s11749-013-0325-7

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