Abstract
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 rth 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.
Highlights
Let T be a linear functional defined over a certain Hilbert space H
H, where L is some centered Gaussian isonormal process, which means that L maps isometrically H onto some Gaussian subspace of L2 (Ω)
Our main goal will be developing procedures which adapt to the smoothness of the underlying function s ∈ H in the framework of model selection as proposed by Barron et al [1] and Birge and Massart [3]
Summary
Let T be a linear functional defined over a certain Hilbert space H. H, where L is some centered Gaussian isonormal process, which means that L maps isometrically H onto some Gaussian subspace of L2 (Ω).
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