Exact adaptive pointwise estimation on Sobolev classes of densities

Butucea, Cristina

Butucea, Cristina

ESAIM: Probability and Statistics, Tome 5 (2001), pp. 1-31.

Résumé

The subject of this paper is to estimate adaptively the common probability density of $n$ independent, identically distributed random variables. The estimation is done at a fixed point $x_{0} \in ℝ$ , over the density functions that belong to the Sobolev class $W_{n} (β, L)$ . We consider the adaptive problem setup, where the regularity parameter $β$ is unknown and varies in a given set $B_{n}$ . A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found.

MR Zbl | 1 citation dans Numdam

Classification : 62N01, 62N02, 62G20
Mots clés : density estimation, exact asymptotics, pointwise risk, sharp adaptive estimator

@article{PS_2001__5__1_0,
     author = {Butucea, Cristina},
     title = {Exact adaptive pointwise estimation on {Sobolev} classes of densities},
     journal = {ESAIM: Probability and Statistics},
     pages = {1--31},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2001},
     mrnumber = {1845320},
     zbl = {0990.62032},
     language = {en},
     url = {http://www.numdam.org/item/PS_2001__5__1_0/}
}

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%0 Journal Article
%A Butucea, Cristina
%T Exact adaptive pointwise estimation on Sobolev classes of densities
%J ESAIM: Probability and Statistics
%D 2001
%P 1-31
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Butucea, Cristina. Exact adaptive pointwise estimation on Sobolev classes of densities. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 1-31. http://www.numdam.org/item/PS_2001__5__1_0/

Bibliographie
Cité par

[1] A. Barron, L. Birge and P. Massart, Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1995) 301-413. | MR | Zbl

[2] O.V. Besov, V.L. Il'In and S.M. Nikol'Skii, Integral representations of functions and imbedding theorems. J. Wiley, New York (1978).

[3] L. Birge and P. Massart, From model selection to adaptive estimation, Festschrift fur Lucien Le Cam. Springer (1997) 55-87. | MR | Zbl

[4] L.D. Brown and M.G. Low, A constrained risk inequality with application to nonparametric functional estimation. Ann. Statist. 24 (1996) 2524-2535. | MR | Zbl

[5] C. Butucea, The adaptive rates of convergence in a problem of pointwise density estimation. Statist. Probab. Lett. 47 (2000) 85-90. | MR | Zbl

[6] C. Butucea, Numerical results concerning a sharp adaptive density estimator. Comput. Statist. 1 (2001). | MR | Zbl

[7] L. Devroye and G. Lugosi, A universally acceptable smoothing factor for kernel density estimates. Ann. Statist. 24 (1996) 2499-2512. | MR | Zbl

[8] 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. | MR | Zbl

[9] D.L. Donoho, I. Johnstone, G. Kerkyacharian and D. Picard, Density estimation by wavelet thresholding. Ann. Statist. 24 (1996) 508-539. | MR | Zbl

[10] D.L. Donoho and M.G. Low, Renormalization exponents and optimal pointwise rates of convergence. Ann. Statist. 20 (1992) 944-970. | MR | Zbl

[11] S.Yu. Efromovich, Nonparametric estimation of a density with unknown smoothness. Theory Probab. Appl. 30 (1985) 557-568. | Zbl

[12] S.Yu. Efromovich and M.S. Pinsker, An adaptive algorithm of nonparametric filtering. Automat. Remote Control 11 (1984) 1434-1440. | Zbl

[13] A. Goldenshluger and A. Nemirovski, On spatially adaptive estimation of nonparametric regression. Math. Methods Statist. 6 (1997) 135-170. | MR | Zbl

[14] G.K. Golubev, Adaptive asymptotically minimax estimates of smooth signals. Problems Inform. Transmission 23 (1987) 57-67. | MR | Zbl

[15] G.K. Golubev, Quasilinear estimates for signals in $𝕃_{2}$ . Problems Inform. Transmission 26 (1990) 15-20. | MR | Zbl

[16] G.K. Golubev, Nonparametric estimation of smooth probability densities in $𝕃_{2}$ . Problems Inform. Transmission 28 (1992) 44-54. | MR | Zbl

[17] G.K. Golubev and M. Nussbaum, Adaptive spline estimates in a nonparametric regression model. Theory Probab. Appl. 37 (1992) 521-529. | MR | Zbl

[18] I.A. Ibragimov and R.Z. Hasminskii, Statistical estimation: Asymptotic theory. Springer-Verlag, New York (1981). | MR | Zbl

[19] A. Juditsky, Wavelet estimators: Adapting to unknown smoothness. Math. Methods Statist. 6 (1997) 1-25. | MR | Zbl

[20] G. Kerkyacharian and D. Picard, Density estimation by kernel and wavelet method, optimality in Besov space. Statist. Probab. Lett. 18 (1993) 327-336. | MR | Zbl

[21] G. Kerkyacharian, D. Picard and K. Tribouley, $𝕃_{p}$ adaptive density estimation. Bernoulli 2 (1996) 229-247. | MR | Zbl

[22] J. Klemelä and A.B. Tsybakov, Sharp adaptive estimation of linear functionals, Prépublication 540. LPMA Paris 6 (1999). | Zbl

[23] O.V. Lepskii, On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 (1990) 454-466. | MR | Zbl

[24] O.V. Lepskii, Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 (1991) 682-697. | MR | Zbl

[25] O.V. Lepskii, On problems of adaptive estimation in white Gaussian noise. Advances in Soviet Math. Amer. Math. Soc. 12 (1992b) 87-106. | MR | Zbl

[26] O.V. Lepski and B.Y. Levit, Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 (1998) 123-156. | MR | Zbl

[27] O.V. Lepski, E. Mammen and V.G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 (1997) 929-947. | MR | Zbl

[28] O.V. Lepski and V.G. Spokoiny, Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 (1997) 2512-2546. | MR | Zbl

[29] D. Pollard, Convergence of Stochastic Processes. Springer-Verlag, New York (1984). | MR | Zbl

[30] A.B. Tsybakov, Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist. 26 (1998) 2420-2469. | MR | Zbl

[31] S. Van De Geer, A maximal inequality for empirical processes, Technical Report TW 9505. University of Leiden, Leiden (1995).