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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
KernSmooth
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