The Annals of Statistics

Möbius deconvolution on the hyperbolic plane with application to impedance density estimation

Stephan F. Huckemann, Peter T. Kim, Ja-Yong Koo, and Axel Munk

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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.

Article information

Source
Ann. Statist., Volume 38, Number 4 (2010), 2465-2498.

Dates
First available in Project Euclid: 11 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1278861254

Digital Object Identifier
doi:10.1214/09-AOS783

Mathematical Reviews number (MathSciNet)
MR2676895

Zentralblatt MATH identifier
1203.62055

Subjects
Primary: 62G07: Density estimation
Secondary: 43A80: Analysis on other specific Lie groups [See also 22Exx]

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

Citation

Huckemann, Stephan F.; Kim, Peter T.; Koo, Ja-Yong; Munk, Axel. Möbius deconvolution on the hyperbolic plane with application to impedance density estimation. Ann. Statist. 38 (2010), no. 4, 2465--2498. doi:10.1214/09-AOS783. https://projecteuclid.org/euclid.aos/1278861254


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