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

Béatrice Laurent

ESAIM: Probability and Statistics (2010)

  • 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 (2010): 1-18. <http://eudml.org/doc/104331>.

@article{Laurent2010,
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.; efficient estimation},
language = {eng},
month = {3},
pages = {1-18},
publisher = {EDP Sciences},
title = { Adaptive estimation of a quadratic functional of a density by model selection},
url = {http://eudml.org/doc/104331},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Laurent, Béatrice
TI - Adaptive estimation of a quadratic functional of a density by model selection
JO - ESAIM: Probability and Statistics
DA - 2010/3//
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.; efficient estimation
UR - http://eudml.org/doc/104331
ER -

References

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  14. I.A. Ibragimov, A. Nemirovski and R.Z. Hasminskii, Some problems on nonparametric estimation in Gaussian white noise. Theory Probab. Appl.31 (1986) 391–406.  
  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. B. Laurent, Efficient estimation of integral functionals of a density. Ann. Statist.24 (1996) 659–681.  
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  18. B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist.28 (2000) 1302–1338.  

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