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

Published online by Cambridge University Press:  15 November 2005

Béatrice Laurent
Béatrice Laurent*
Affiliation:
INSA-LSP. Departement GMM, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France; beatrice.laurent@insa-toulouse.fr
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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.

Keywords

Adaptive estimationquadratic functionalsmodel selectionBesov bodiesefficient estimation.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 1 - 18
Copyright
© EDP Sciences, SMAI, 2005

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References

Bickel, P. and Ritov, Y., Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhya Ser. A. 50 (1989) 381–393.
Birgé, L. and Massart, P., Estimation of integral functionals of a density. Ann. Statist. 23 (1995) 11–29. CrossRef
Birgé, L. and Massart, P., Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 (1998) 329–375. CrossRef
L. Birgé and Y. Rozenholc, How many bins should be put in a regular histogram. Technical Report Université Paris 6 et 7 (2002).
Bretagnolle, J., A new large deviation inequality for U-statistics of order 2. ESAIM: PS 3 (1999) 151–162. CrossRef
Donoho, D. and Nussbaum, M., Minimax quadratic estimation of a quadratic functional. J. Complexity 6 (1990) 290–323. CrossRef
Efroïmovich, S. and Low, M., Bickel, On and Ritov's conjecture about adaptive estimation of the integral of the square of density derivatives. Ann. Statist. 24 (1996) 682–686.
Efroïmovich, S. and Low, M., 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).
Gayraud, G. and Tribouley, K., Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law. Statist. Probab. Lett. 44 (1999) 109–122. CrossRef
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).
Ibragimov, I.A., Nemirovski, A. and Hasminskii, R.Z., Some problems on nonparametric estimation in Gaussian white noise. Theory Probab. Appl. 31 (1986) 391–406. CrossRef
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.
Laurent, B., 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. Bernoulli 3 (1997) 181–211.
Laurent, B. and Massart, P., Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 (2000) 1302–1338.