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Asymptotics for spectral regularization estimators in statistical inverse problems

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  • Nicolai Bissantz
  • Hajo Holzmann

Abstract

While optimal rates of convergence in L 2 for spectral regularization estimators in statistical inverse problems have been much studied, the pointwise asymptotics for these estimators have received very little consideration. Here, we briefly discuss asymptotic expressions for bias and variance for some such estimators, mainly in deconvolution-type problems, and also show their asymptotic normality. The main part of the paper consists of a simulation study in which we investigate in detail the pointwise finite sample properties, both for deconvolution and the backward heat equation as well as for a regression model involving the Radon transform. In particular we explore the practical use of undersmoothing in order to achieve the nominal coverage probabilities of the confidence intervals. Copyright Springer-Verlag 2013

Suggested Citation

  • Nicolai Bissantz & Hajo Holzmann, 2013. "Asymptotics for spectral regularization estimators in statistical inverse problems," Computational Statistics, Springer, vol. 28(2), pages 435-453, April.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:2:p:435-453
    DOI: 10.1007/s00180-012-0309-1
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    References listed on IDEAS

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    1. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    2. Iain M. Johnstone & Gérard Kerkyacharian & Dominique Picard & Marc Raimondo, 2004. "Wavelet deconvolution in a periodic setting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 547-573, August.
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