Adaptive estimation of linear functionals by model selection

Details Adaptive estimation of linear functionals by model selection Adaptive estimation of linear functionals by model selection

2007-10-09

arxiv.org/abs/0710.1688v2

Electronic Journal of Statistics 2008, Vol. 2, No. 0, 993-1020

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/07-EJS127 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t

Scientific paper

10.1214/07-EJS127

We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\mathbb{L}_p$ loss. An application to the problem of estimating a signal or its $r^{th}$ derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. We also apply our non asymptotic oracle inequality to the estimation of the mean of the signal on an interval with length depending on the noise level. Simulations are included to illustrate the performances of the procedure for the estimation of a function at a given point. Our method provides a pointwise adaptive estimator.

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