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Abstract Inverse Estimation with Application to Deconvolution on Locally Compact Abelian Groups

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Abstract

Recovery of the unknown parameter in an abstract inverse estimation model can be based on regularizing the inverse of the operator defining the model. Such regularized-inverse type estimators are constructed with the help of a version of the spectral theorem due to Halmos, after suitable preconditioning. A lower bound to the minimax risk is obtained exploiting the van Trees inequality. The proposed estimators are shown to be asymptotically optimal in the sense that their risk converges to zero, as the sample size tends to infinity, at the same rate as this lower bound. The general theory is applied to deconvolution on locally compact Abelian groups, including both indirect density and indirect regression function estimation.

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Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Nijmegen, 6525, ED Nijmegen, The Netherlands

    Arnoud C.M. van Rooij

  2. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA

    Frits H. Ruymgaart

Authors
  1. Arnoud C.M. van Rooij
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  2. Frits H. Ruymgaart
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van Rooij, A.C., Ruymgaart, F.H. Abstract Inverse Estimation with Application to Deconvolution on Locally Compact Abelian Groups. Annals of the Institute of Statistical Mathematics 53, 781–798 (2001). https://doi.org/10.1023/A:1014665305349

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