ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, February 2005, Vol. 9, pp. 1-18
DOI: 10.1051/ps:2005001

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

Béatrice Laurent

INSA-LSP. Departement GMM, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France; beatrice.laurent@insa-toulouse.fr

(Received October 23, 2003. Revised July 16, 2004.)

Abstract
We consider the problem of estimating the integral of the square of a density f from the observation of a n sample. Our method to estimate $\int_{\mathbb f^2(x){\rm d}x$ is based 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 exponential inequality for U-statistics of order 2 due to Houdré and Reynaud.

Mathematics Subject Classification. 62G05, 62G20, 62J02.

Key words: Adaptive estimation, quadratic functionals, model selection, Besov bodies, efficient estimation.

© EDP Sciences, SMAI 2005