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On adaptive inverse estimation of linear functional in Hilbert scales. (English) Zbl 1055.62034
Summary: We address the problem of estimating the value of a linear functional $$\langle f,x\rangle$$ 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.

##### MSC:
 62G07 Density estimation 46N30 Applications of functional analysis in probability theory and statistics 62G05 Nonparametric estimation
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