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

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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 f 2 (x)dx 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 : 10.1051/ps:2005001
Classification : 62G05, 62G20, 62J02
Keywords: adaptive estimation, quadratic functionals, model selection, Besov bodies, efficient estimation
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     author = {Laurent, B\'eatrice},
     title = {Adaptive estimation of a quadratic functional of a density by model selection},
     journal = {ESAIM: Probability and Statistics},
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     publisher = {EDP-Sciences},
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     year = {2005},
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     zbl = {1136.62333},
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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

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