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August 2010 Möbius deconvolution on the hyperbolic plane with application to impedance density estimation
Stephan F. Huckemann, Peter T. Kim, Ja-Yong Koo, Axel Munk
Ann. Statist. 38(4): 2465-2498 (August 2010). DOI: 10.1214/09-AOS783


In this paper we consider a novel statistical inverse problem on the Poincaré, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 × 2 real matrices of determinant one via Möbius transformations. Our approach is based on a deconvolution technique which relies on the Helgason–Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random Möbius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincaré plane exactly describes the physical system that is of statistical interest.


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Stephan F. Huckemann. Peter T. Kim. Ja-Yong Koo. Axel Munk. "Möbius deconvolution on the hyperbolic plane with application to impedance density estimation." Ann. Statist. 38 (4) 2465 - 2498, August 2010.


Published: August 2010
First available in Project Euclid: 11 July 2010

zbMATH: 1203.62055
MathSciNet: MR2676895
Digital Object Identifier: 10.1214/09-AOS783

Primary: 62G07
Secondary: 43A80

Keywords: Cayley transform , cross-validation , Deconvolution , Fourier analysis , Helgason–Fourier transform , Hyperbolic space , impedance , Laplace–Beltrami operator , Möbius transformation , special linear group , Statistical inverse problems , upper half-plane

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 4 • August 2010
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