## Electronic Journal of Statistics

### Adaptive estimation of linear functionals by model selection

#### 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.

#### Article information

Source
Electron. J. Statist., Volume 2 (2008), 993-1020.

Dates
First available in Project Euclid: 27 October 2008

https://projecteuclid.org/euclid.ejs/1225114040

Digital Object Identifier
doi:10.1214/07-EJS127

Mathematical Reviews number (MathSciNet)
MR2448602

Zentralblatt MATH identifier
1320.62074

Subjects
Primary: 62G05: Estimation 62G08: Nonparametric regression

#### Citation

Laurent, Béatrice; Ludeña, Carenne; Prieur, Clémentine. Adaptive estimation of linear functionals by model selection. Electron. J. Statist. 2 (2008), 993--1020. doi:10.1214/07-EJS127. https://projecteuclid.org/euclid.ejs/1225114040

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