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Möbius deconvolution on the hyperbolic plane with application to impedance density estimation. (English) Zbl 1203.62055
Summary: 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 \(2 \times 2\) 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.

MSC:
62G07 Density estimation
43A80 Analysis on other specific Lie groups
62P35 Applications of statistics to physics
62P30 Applications of statistics in engineering and industry; control charts
43A99 Abstract harmonic analysis
Software:
KernSmooth
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