Opuscula Math. 36, no. 5 (2016), 671-679

Opuscula Mathematica

On a dense minimizer of empirical risk in inverse problems

Jacek Podlewski
Zbigniew Szkutnik

Abstract. Properties of estimators of a functional parameter in an inverse problem setup are studied. We focus on estimators obtained through dense minimization (as opposed to minimization over \(\delta\)-nets) of suitably defined empirical risk. At the cost of imposition of a sort of local finite-dimensionality assumption, we fill some gaps in the proofs of results published by Klemelä and Mammen [Ann. Statist. 38 (2010), 482-511]. We also give examples of functional classes that satisfy the modified assumptions.

Keywords: inverse problem, empirical risk minimization.

Mathematics Subject Classification: 62G07.

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  • Jacek Podlewski
  • StatSoft Poland, ul. Kraszewskiego 36, 30-110 Krakow, Poland
  • Zbigniew Szkutnik
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
  • Communicated by Andrzej Kozek.
  • Received: 2015-10-15.
  • Revised: 2016-03-10.
  • Accepted: 2016-03-10.
  • Published online: 2016-06-29.
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Cite this article as:
Jacek Podlewski, Zbigniew Szkutnik, On a dense minimizer of empirical risk in inverse problems, Opuscula Math. 36, no. 5 (2016), 671-679, http://dx.doi.org/10.7494/OpMath.2016.36.5.671

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