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

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### abstract

• In this paper we consider a novel statistical inverse problem on the Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of $2\times2$ real matrices of determinant one via M\"{o}bius 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 M\"{o}bius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincar\'{e} plane exactly describes the physical system that is of statistical interest.

### authors

• Huckemann, Stephan F
• Kim, Peter
• Koo, Ja-Yong
• Munk, Axel

• August 2010