Thresholding projection estimators in functional linear models

Authors:
Hervé Cardot;Jan Johannes
Affiliations:
Université de Bourgogne, Institut de Mathématiques de Bourgogne, 9 Av. Alain Savary, 21078 Dijon Cedex, France;Universität Heidelberg, Institut für Angewandte Mathematik, Im Neuenheimer Feld, 294, D-69120 Heidelberg, Germany
Venue:
Journal of Multivariate Analysis
Year:
2010

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Abstract

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows us to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits us to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove that these estimators are minimax and rates of convergence are given for some particular cases.