On adaptive inverse estimating linear functionals of unknown smoothness in Hilbert scales
- Goldenshluger, Alexander
- Pereverzev, Sergei V.
2010 Mathematics Subject Classification
- 62G05 62G20 65R30
- adaptive estimation, Hilbert scales, inverse problems, linear functionals, minimax risk, regularization
We address the problem of estimating the value of a linear functional 〈 ƒ,𝑥 〉 from random noisy observations of 𝑦 = A 𝑥 in Hilbert scales. Both the white noise and density observation models are considered. We develop an inverse estimator of 〈 ƒ,𝑥 〉 which automatically adapts to unknown smoothness of 𝑥 and ƒ. It is shown that accuracy of this adaptive estimator is only by a logarithmic factor worse 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.