Abstract.
We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale (that includes ellipoids with polynomially and exponentially decreasing axes) and, at the same time, satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non-increasing weights. The construction of the estimator is based on a properly penalized blockwise Stein's rule, with weakly geometically increasing blocks. As an application, we construct sharp adaptive estimators in the problems of deconvolution and tomography.
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Received: 19 January 2000 / Revised version: 30 April 2001 / Published online: 14 June 2002
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Cavalier, L., Tsybakov, A. Sharp adaptation for inverse problems with random noise. Probab Theory Relat Fields 123, 323–354 (2002). https://doi.org/10.1007/s004400100169
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DOI: https://doi.org/10.1007/s004400100169