Adaptive estimation of a quadratic functional of a density by model selection
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 1-18.

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 ${\int }_{ℝ}{f}^{2}\left(x\right)\mathrm{d}x$ 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.

DOI : https://doi.org/10.1051/ps:2005001
Classification : 62G05,  62G20,  62J02
Mots clés : adaptive estimation, quadratic functionals, model selection, Besov bodies, efficient estimation
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Laurent, Béatrice. Adaptive estimation of a quadratic functional of a density by model selection. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 1-18. doi : 10.1051/ps:2005001. http://www.numdam.org/articles/10.1051/ps:2005001/

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