2011/07/07 by Johannes Schmidt-Hieber, Schmidt-Hieber, Johannes, Axel Munk +3 · 1 citation
Computer Science · Engineering · Materials Science · Mathematics · #62G10 (Primary) 62G15 #62G20 (Secondary) #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Machine Learning in Materials Science #Methodology (stat.ME) #Reservoir Engineering and Simulation Methods #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:62G10 #msc:62G15 #msc:62G20 #stat.ME #stat.TH
paper · pdf · doi:10.48550/arxiv.1107.1404
55 pages, 5 figures, This is a revised version of a previous paper with the title: "Multiscale Methods for Shape Constraints in Deconvolution"
openalex publication_date 2011/07/07 · arxiv created 2012/12/17 · arxiv updated 2015/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow.