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June 2013 Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features
Johannes Schmidt-Hieber, Axel Munk, Lutz Dümbgen
Ann. Statist. 41(3): 1299-1328 (June 2013). DOI: 10.1214/13-AOS1089

Abstract

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.

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Johannes Schmidt-Hieber. Axel Munk. Lutz Dümbgen. "Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features." Ann. Statist. 41 (3) 1299 - 1328, June 2013. https://doi.org/10.1214/13-AOS1089

Information

Published: June 2013
First available in Project Euclid: 4 July 2013

zbMATH: 1293.62104
MathSciNet: MR3113812
Digital Object Identifier: 10.1214/13-AOS1089

Subjects:
Primary: 62G10
Secondary: 62G15 , 62G20

Keywords: Brownian motion , convexity , Ill-posed problems , mode detection , Monotonicity , multiscale statistics , pseudo-differential operators , shape constraints

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3 • June 2013
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