The Annals of Statistics

Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes

A. B. Tsybakov

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The problem of nonparametric function estimation in the Gaussian white noise model is considered. It is assumed that the unknown function belongs to one of the Sobolev classes, with an unknown regularity parameter. Asymptotically exact adaptive estimators of functions are proposed on the scale of Sobolev classes, with respect to pointwise and sup-norm risks. It is shown that, unlike the case of $L_2$-risk, a loss of efficiency under adaptation is inevitable here. Bounds on the value of the loss of efficiency are obtained.

Article information

Ann. Statist., Volume 26, Number 6 (1998), 2420-2469.

First available in Project Euclid: 21 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Gaussian white noise adaptive nonparametric estimation Sobolev class loss of efficiency under adaptation minimax risk exact constants


Tsybakov, A. B. Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist. 26 (1998), no. 6, 2420--2469. doi:10.1214/aos/1024691478.

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