Mair, Bernard A.; Ruymgaart, Frits H. Statistical inverse estimation in Hilbert scales. (English) Zbl 0864.62020 SIAM J. Appl. Math. 56, No. 5, 1424-1444 (1996). Summary: The recover of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is available. For greater flexibility the general problem is embedded in an abstract Hilbert scale. In the applications Sobolev scales are used. For the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product, where the operator is bounded but not necessarily compact. A lower bound to certain minimax rates is included, and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate. Examples include errors-in-variables (deconvolution) and indirect nonparametric regression. Special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation. Cited in 89 Documents MSC: 62G07 Density estimation 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 46N30 Applications of functional analysis in probability theory and statistics Keywords:inverse estimation; deconvolution; recover of signals; indirect measurements; random noise; smoothness; Hilbert scale; Sobolev scales; regularized operator inversion; asymptotic minimax rate; errors-in-variables; indirect nonparametric regression; heat equation PDFBibTeX XMLCite \textit{B. A. Mair} and \textit{F. H. Ruymgaart}, SIAM J. Appl. Math. 56, No. 5, 1424--1444 (1996; Zbl 0864.62020) Full Text: DOI