×

Sharp adaptation for inverse problems with random noise. (English) Zbl 1039.62031

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.

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
PDFBibTeX XMLCite
Full Text: DOI