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MOBIUS DECONVOLUTION ON THE HYPERBOLIC PLANE WITH APPLICATION TO IMPEDANCE DENSITY ESTIMATION

Authors
Huckemann, Stephan F.Kim, Peter T.Koo, Ja-YongMunk, Axel
Issue Date
8월-2010
Publisher
INST MATHEMATICAL STATISTICS
Keywords
Cayley transform; cross-validation; deconvolution; Fourier analysis; Helgason-Fourier transform; hyperbolic space; impedance; Laplace-Beltrami operator; Mobius transformation; special linear group; statistical inverse problems; upper half-plane
Citation
ANNALS OF STATISTICS, v.38, no.4, pp.2465 - 2498
Indexed
SCIE
SCOPUS
Journal Title
ANNALS OF STATISTICS
Volume
38
Number
4
Start Page
2465
End Page
2498
URI
https://scholar.korea.ac.kr/handle/2021.sw.korea/116022
DOI
10.1214/09-AOS783
ISSN
0090-5364
Abstract
In this paper we consider a novel statistical inverse problem on the Poincare, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via Mobius 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 Mains transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincare plane exactly describes the physical system that is of statistical interest.
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