Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. (English) Zbl 1055.62523
Summary: 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.
MSC:
62G05 | Nonparametric estimation |
46N30 | Applications of functional analysis in probability theory and statistics |
62G20 | Asymptotic properties of nonparametric inference |
65R30 | Numerical methods for ill-posed problems for integral equations |