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

ESAIM: Probability and Statistics (2010)

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

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topLaurent, 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

top- P. Bickel and Y. Ritov, Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhya Ser. A.50 (1989) 381–393.
- L. Birgé and P. Massart, Estimation of integral functionals of a density. Ann. Statist.23 (1995) 11–29.
- L. Birgé and P. Massart, Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli4 (1998) 329–375.
- L. Birgé and Y. Rozenholc, How many bins should be put in a regular histogram. Technical Report Université Paris 6 et 7 (2002).
- J. Bretagnolle, A new large deviation inequality for U-statistics of order 2. ESAIM: PS 3 (1999) 151–162.
- D. Donoho and M. Nussbaum, Minimax quadratic estimation of a quadratic functional. J. Complexity6 (1990) 290–323.
- S. Efroïmovich and M. Low, On Bickel and Ritov's conjecture about adaptive estimation of the integral of the square of density derivatives. Ann. Statist.24 (1996) 682–686.
- S. Efroïmovich and M. Low, On optimal adaptive estimation of a quadratic functional. Ann. Statist.24 (1996) 1106–1125.
- M. Fromont and B. Laurent, Adaptive goodness-of-fit tests in a density model. Technical report. Université Paris 11 (2003).
- G. Gayraud and K. Tribouley, Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law. Statist. Probab. Lett.44 (1999) 109–122.
- E. Giné, R. Latala and J. Zinn, Exponential and moment inequalities for U-statistics. High Dimensional Probability 2, Progress in Probability 47 (2000) 13–38.
- W. Hardle, G. Kerkyacharian, D. Picard, A. Tsybakov, Wavelets, Approximations and statistical applications. Lect. Notes Stat.129 (1998).
- 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).
- 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.
- 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.
- B. Laurent, Efficient estimation of integral functionals of a density. Ann. Statist.24 (1996) 659–681.
- B. Laurent, Estimation of integral functionals of a density and its derivatives. Bernoulli3 (1997) 181–211.
- B. Laurent and P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Statist.28 (2000) 1302–1338.

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