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

• 成果类型: Article
• 署名作者: Huckemann, Stephan F.; Kim, Peter T.; Koo, Ja-Yong; Munk, Axel
• 署名单位: University of Gottingen; University of Guelph; Korea University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS783
• 发表日期: 2010
• 页码: 2465-2498
• 关键词: statistical inverse problems; extrinsic sample means; parameter; selection; regularization; CONVERGENCE; MANIFOLDS; rates

摘要

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.