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

Béatrice Laurent

ESAIM: Probability and Statistics (2005)

  • Volume: 9, page 1-18
  • ISSN: 1292-8100

Abstract

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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 ∫ ℝ f 2 ( x ) 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.

How to cite

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Laurent, Béatrice. "Adaptive estimation of a quadratic functional of a density by model selection." ESAIM: Probability and Statistics 9 (2005): 1-18. <http://eudml.org/doc/245507>.

@article{Laurent2005,
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.},
author = {Laurent, Béatrice},
journal = {ESAIM: Probability and Statistics},
keywords = {adaptive estimation; quadratic functionals; model selection; Besov bodies; efficient estimation; Adaptive estimation},
language = {eng},
pages = {1-18},
publisher = {EDP-Sciences},
title = {Adaptive estimation of a quadratic functional of a density by model selection},
url = {http://eudml.org/doc/245507},
volume = {9},
year = {2005},
}

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
PB - EDP-Sciences
VL - 9
SP - 1
EP - 18
AB - 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.
LA - eng
KW - adaptive estimation; quadratic functionals; model selection; Besov bodies; efficient estimation; Adaptive estimation
UR - http://eudml.org/doc/245507
ER -

References

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  13. [13] C. Houdré and P. Reynaud-Bouret, Exponential inequalities for U-statistics of order two with constants, in Euroconference on Stochastic inequalities and applications. Barcelona. Birkhauser (2002). Zbl1036.60015
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  15. [15] I. Johnstone, Chi-square oracle inequalities. State of the art in probability and statistics (Leiden 1999) - IMS Lecture Notes Monogr. Ser., 36. Inst. Math. Statist., Beachwood, OH (1999) 399–418. 
  16. [16] B. Laurent, Efficient estimation of integral functionals of a density. Ann. Statist. 24 (1996) 659–681. Zbl0859.62038
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  18. [18] B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 (2000) 1302–1338. Zbl1105.62328

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