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Exact adaptive pointwise estimationon Sobolev classes of densities

Published online by Cambridge University Press:  15 August 2002

Cristina Butucea
Cristina Butucea*
Affiliation:
Université Paris 10, Modal'X, bâtiment G, 200 avenue de la République, 92001 Nanterre, France; cbutucea@u-paris10.fr. and Université Paris 6, Laboratoire Probabilités et Modèles Aléatoires, 6 rue Clisson, 75013 Paris, France; butucea@ccr.jussieu.fr.
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Abstract

The subject of this paper is to estimate adaptively the common probabilitydensity of n independent, identically distributed random variables. Theestimation is done at a fixed point $x_{0}\in \mathbb R$ , over the densityfunctions that belong to the Sobolev class Wn(β,L). We consider theadaptive problem setup, where the regularity parameter β is unknownand varies in a given set B n . A sharp adaptive estimator is obtained,and the explicit asymptotical constant, associated to its rate ofconvergence is found.

Keywords

Density estimationexact asymptoticspointwise risksharp adaptive estimator.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 5 , 2001 , pp. 1 - 31
Copyright
© EDP Sciences, SMAI, 2001

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References

Barron, A., Birge, L. and Massart, P., Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1995) 301-413. CrossRef
O.V. Besov, V.L. Il'in and S.M. Nikol'skii, Integral representations of functions and imbedding theorems. J. Wiley, New York (1978).
L. Birge and P. Massart, From model selection to adaptive estimation, Festschrift fur Lucien Le Cam. Springer (1997) 55-87.
Brown, L.D. and Low, M.G., A constrained risk inequality with application to nonparametric functional estimation. Ann. Statist. 24 (1996) 2524-2535.
Butucea, C., The adaptive rates of convergence in a problem of pointwise density estimation. Statist. Probab. Lett. 47 (2000) 85-90. CrossRef
C. Butucea, Numerical results concerning a sharp adaptive density estimator. Comput. Statist. 1 (2001).
Devroye, L. and Lugosi, G., A universally acceptable smoothing factor for kernel density estimates. Ann. Statist. 24 (1996) 2499-2512.
D.L. Donoho, I. Johnstone, G. Kerkyacharian and D. Picard, Wavelet shrinkage: Asymptopia? J. R. Stat. Soc. Ser. B Stat. Methodol. 57 (1995) 301-369.
Donoho, D.L., Johnstone, I., Kerkyacharian, G. and Picard, D., Density estimation by wavelet thresholding. Ann. Statist. 24 (1996) 508-539.
Donoho, D.L. and Low, M.G., Renormalization exponents and optimal pointwise rates of convergence. Ann. Statist. 20 (1992) 944-970. CrossRef
Efromovich, S.Yu., Nonparametric estimation of a density with unknown smoothness. Theory Probab. Appl. 30 (1985) 557-568. CrossRef
Efromovich, S.Yu. and Pinsker, M.S., An adaptive algorithm of nonparametric filtering. Automat. Remote Control 11 (1984) 1434-1440.
Goldenshluger, A. and Nemirovski, A., On spatially adaptive estimation of nonparametric regression. Math. Methods Statist. 6 (1997) 135-170.
Golubev, G.K., Adaptive asymptotically minimax estimates of smooth signals. Problems Inform. Transmission 23 (1987) 57-67.
Golubev, G.K., Quasilinear estimates for signals in $\mathbb{L}_{2}$ . Problems Inform. Transmission 26 (1990) 15-20.
Golubev, G.K., Nonparametric estimation of smooth probability densities in $\mathbb{L}_{2}$ . Problems Inform. Transmission 28 (1992) 44-54.
Golubev, G.K. and Nussbaum, M., Adaptive spline estimates in a nonparametric regression model. Theory Probab. Appl. 37 (1992) 521-529. CrossRef
I.A. Ibragimov and R.Z. Hasminskii, Statistical estimation: Asymptotic theory. Springer-Verlag, New York (1981).
Juditsky, A., Wavelet estimators: Adapting to unknown smoothness. Math. Methods Statist. 6 (1997) 1-25.
Kerkyacharian, G. and Picard, D., Density estimation by kernel and wavelet method, optimality in Besov space. Statist. Probab. Lett. 18 (1993) 327-336. CrossRef
Kerkyacharian, G., Picard, D. and Tribouley, K., $\mathbb{L}_{p}$ adaptive density estimation. Bernoulli 2 (1996) 229-247.
J. Klemelä and A.B. Tsybakov, Sharp adaptive estimation of linear functionals, Prépublication 540. LPMA Paris 6 (1999).
Lepskii, O.V., On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 (1990) 454-466. CrossRef
Lepskii, O.V., Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 (1991) 682-697. CrossRef
Lepskii, O.V., On problems of adaptive estimation in white Gaussian noise. Advances in Soviet Math. Amer. Math. Soc. 12 (1992b) 87-106.
Lepski, O.V. and Levit, B.Y., Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 (1998) 123-156.
Lepski, O.V., Mammen, E. and Spokoiny, V.G., Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 (1997) 929-947.
Lepski, O.V. and Spokoiny, V.G., Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 (1997) 2512-2546.
D. Pollard, Convergence of Stochastic Processes. Springer-Verlag, New York (1984).
Tsybakov, A.B., Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist. 26 (1998) 2420-2469.
S. Van de Geer, A maximal inequality for empirical processes, Technical Report TW 9505. University of Leiden, Leiden (1995).