Electronic Journal of Statistics

Estimation and detection of functions from anisotropic Sobolev classes

Yuri Ingster and Natalia Stepanova

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Abstract

We consider the problems of estimating and detecting an unknown function f depending on a multidimensional variable (for instance, an image) observed in the Gaussian white noise. It is assumed that f belongs to anisotropic Sobolev class. The case of a function of infinitely many variables is also considered. An asymptotic study (as the noise level tends to zero) of the estimation and detection problems is done. In connection with the estimation problem, we construct asymptotically minimax estimators and establish sharp asymptotics for the minimax integrated squared risk. In the detection problem, we construct asymptotically minimax tests and provide conditions for distinguishability in the problem.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 484-506.

Dates
First available in Project Euclid: 2 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1307023008

Digital Object Identifier
doi:10.1214/11-EJS615

Mathematical Reviews number (MathSciNet)
MR2813552

Zentralblatt MATH identifier
1274.62319

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Keywords
Nonparametric estimation nonparametric signal detection Gaussian white noise multivariate functions anisotropic smoothness

Citation

Ingster, Yuri; Stepanova, Natalia. Estimation and detection of functions from anisotropic Sobolev classes. Electron. J. Statist. 5 (2011), 484--506. doi:10.1214/11-EJS615. https://projecteuclid.org/euclid.ejs/1307023008


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