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Adaptive estimation of a quadratic functional of a density by model selection

Published online by Cambridge University Press:  15 November 2005

Béatrice Laurent
Béatrice Laurent*
Affiliation:
INSA-LSP. Departement GMM, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France; beatrice.laurent@insa-toulouse.fr
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Abstract

We consider the problem of estimating the integral of the square of a densityf from the observation of a n sample. Our method to estimate $\int_{\mathbb{R}} f^2(x){\rm d}x$ isbased on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponentialinequality for U-statistics of order 2 due to Houdré and Reynaud.

Keywords

Adaptive estimationquadratic functionalsmodel selectionBesov bodiesefficient estimation.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 1 - 18
Copyright
© EDP Sciences, SMAI, 2005

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