The Annals of Statistics

Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features

Johannes Schmidt-Hieber, Axel Munk, and Lutz Dümbgen

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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.

Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1299-1328.

Dates
First available in Project Euclid: 4 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1372979639

Digital Object Identifier
doi:10.1214/13-AOS1089

Mathematical Reviews number (MathSciNet)
MR3113812

Zentralblatt MATH identifier
1293.62104

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

Keywords
Brownian motion convexity pseudo-differential operators ill-posed problems mode detection monotonicity multiscale statistics shape constraints

Citation

Schmidt-Hieber, Johannes; Munk, Axel; Dümbgen, Lutz. Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features. Ann. Statist. 41 (2013), no. 3, 1299--1328. doi:10.1214/13-AOS1089. https://projecteuclid.org/euclid.aos/1372979639


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Supplemental materials

  • Supplementary material: Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features. All proofs can be found in the supplementary part, which contains additionally various lemmas, enumerated by $B.1,B.2,\ldots,C.1,C.2,\ldots.$.