M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation
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 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.
Cite
@article{arxiv.1010.4202,
title = {M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation},
author = {Stephan F. Huckemann and Peter T. Kim and Ja-Yong Koo and Axel Munk},
journal= {arXiv preprint arXiv:1010.4202},
year = {2010}
}
Comments
Published in at http://dx.doi.org/10.1214/09-AOS783 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)