Open Access
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

Abstract

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.

Citation

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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. https://doi.org/10.1214/09-AOS783

Information

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

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

Subjects:
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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