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October 2003 On adaptive inverse estimation of linear functionals in Hilbert scales
Alexander Goldenshluger, Sergei V. Pereverzev
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Bernoulli 9(5): 783-807 (October 2003). DOI: 10.3150/bj/1066418878


We address the problem of estimating the value of a linear functional $local asymptotic normalitygle f,x \rangle$ from random noisy observations of $y=Ax$ in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of $x$, of $f$, and of the noise covariance operator. It is shown that accuracy of this adaptive estimator is worse only by a logarithmic factor than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.


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Alexander Goldenshluger. Sergei V. Pereverzev. "On adaptive inverse estimation of linear functionals in Hilbert scales." Bernoulli 9 (5) 783 - 807, October 2003.


Published: October 2003
First available in Project Euclid: 17 October 2003

zbMATH: 1055.62034
MathSciNet: MR2047686
Digital Object Identifier: 10.3150/bj/1066418878

Keywords: adaptive estimation , Hilbert scales , Inverse problems , linear functionals , minimax risk , regularization

Rights: Copyright © 2003 Bernoulli Society for Mathematical Statistics and Probability

Vol.9 • No. 5 • October 2003
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