Open Access
2011 Estimation and detection of functions from anisotropic Sobolev classes
Yuri Ingster, Natalia Stepanova
Electron. J. Statist. 5: 484-506 (2011). DOI: 10.1214/11-EJS615


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.


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Yuri Ingster. Natalia Stepanova. "Estimation and detection of functions from anisotropic Sobolev classes." Electron. J. Statist. 5 484 - 506, 2011.


Published: 2011
First available in Project Euclid: 2 June 2011

zbMATH: 1274.62319
MathSciNet: MR2813552
Digital Object Identifier: 10.1214/11-EJS615

Primary: 62G10
Secondary: 62G20

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

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

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