On Adaptive Inverse Estimation of Linear Functionals in Hilbert Scales
We address the problem of estimating the value of a linear functional 〈f, x〉 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 noi...
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| Veröffentlicht in: | Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2003-10, Vol.9 (5), p.783-807 |
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| container_end_page | 807 |
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| container_issue | 5 |
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| container_title | Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability |
| container_volume | 9 |
| creator | Goldenshluger, Alexander Pereverzev, Sergei V. |
| description | We address the problem of estimating the value of a linear functional 〈f, x〉 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. |
| doi_str_mv | 10.3150/bj/1066418878 |
| format | Article |
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| ispartof | Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability, 2003-10, Vol.9 (5), p.783-807 |
| issn | 1350-7265 |
| language | eng |
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| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| subjects | adaptive estimation Covariance Density estimation Estimate reliability Estimators Hilbert scales Hilbert spaces inverse problems linear functionals Mathematical constants Mathematical theorems Minimax minimax risk Random variables regularization White noise |
| title | On Adaptive Inverse Estimation of Linear Functionals in Hilbert Scales |
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