Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations
- Goldenshluger, Alexander
- Pereverzev, Sergei V.
2010 Mathematics Subject Classification
- 62G05 65J10 41A25
- Adaptive estimation, discretization, Hilbert scales, inverse problems, linear functionals, regularization, minimax risk
We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals. The case of discretized indirect white noise observations is studied, and the adaptive estimator in this setting is developed.