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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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

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

First available in Project Euclid: 4 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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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 $B.1,B.2,\ldots,C.1,C.2,\ldots.$.