## Bernoulli

• Bernoulli
• Volume 9, Number 5 (2003), 783-807.

### On adaptive inverse estimation of linear functionals in Hilbert scales

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 9, Number 5 (2003), 783-807.

Dates
First available in Project Euclid: 17 October 2003

https://projecteuclid.org/euclid.bj/1066418878

Digital Object Identifier
doi:10.3150/bj/1066418878

Mathematical Reviews number (MathSciNet)
MR2047686

Zentralblatt MATH identifier
1055.62034

#### Citation

Goldenshluger, Alexander; Pereverzev, Sergei V. On adaptive inverse estimation of linear functionals in Hilbert scales. Bernoulli 9 (2003), no. 5, 783--807. doi:10.3150/bj/1066418878. https://projecteuclid.org/euclid.bj/1066418878