Cavalier, Laurent; Tsybakov, Alexandre Sharp adaptation for inverse problems with random noise. (English) Zbl 1039.62031 Probab. Theory Relat. Fields 123, No. 3, 323-354 (2002). Summary: 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 geometrically increasing blocks. As an application, we construct sharp adaptive estimators in the problems of deconvolution and tomography. Cited in 66 Documents MSC: 62G07 Density estimation 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62H12 Estimation in multivariate analysis 62C20 Minimax procedures in statistical decision theory Keywords:adaptive curve estimation; statistical inverse problem; exact minimax constants; oracle inequalities; tomography; deconvolution PDFBibTeX XMLCite \textit{L. Cavalier} and \textit{A. Tsybakov}, Probab. Theory Relat. Fields 123, No. 3, 323--354 (2002; Zbl 1039.62031) Full Text: DOI