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

Keywords: adaptive estimation, quadratic functionals, model selection, Besov bodies, efficient estimation

@article{PS_2005__9__1_0, author = {Laurent, B\'eatrice}, title = {Adaptive estimation of a quadratic functional of a density by model selection}, journal = {ESAIM: Probability and Statistics}, pages = {1--18}, publisher = {EDP-Sciences}, volume = {9}, year = {2005}, doi = {10.1051/ps:2005001}, mrnumber = {2148958}, zbl = {1136.62333}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps:2005001/} }

TY - JOUR AU - Laurent, Béatrice TI - Adaptive estimation of a quadratic functional of a density by model selection JO - ESAIM: Probability and Statistics PY - 2005 SP - 1 EP - 18 VL - 9 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2005001/ DO - 10.1051/ps:2005001 LA - en ID - PS_2005__9__1_0 ER -

Laurent, Béatrice. Adaptive estimation of a quadratic functional of a density by model selection. ESAIM: Probability and Statistics, Volume 9 (2005), pp. 1-18. doi : 10.1051/ps:2005001. http://archive.numdam.org/articles/10.1051/ps:2005001/

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