ESAIM: Probability and Statistics

Table of Contents - November 2005 - Volume 9  

Research Article

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

Laurent, Béatrice

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

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 $\int_{\mathbb{R}} f^2(x){\rm d}x$ 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.

(Received October 23 2003)

(Revised July 16 2004)

(Online publication November 15 2005)

Key Words:

  • Adaptive estimation;
  • quadratic functionals;
  • model selection;
  • Besov bodies;
  • efficient estimation.

Mathematics Subject Classification:

  • 62G05;
  • 62G20;
  • 62J02
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