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Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators

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  • Bissantz, Nicolai
  • Birke, Melanie

Abstract

We consider inverse regression models with convolution-type operators which mediate convolution on (d>=1) and prove a pointwise central limit theorem for spectral regularisation estimators which can be applied to construct pointwise confidence regions. Here, we cope with the unknown bias of such estimators by undersmoothing. Moreover, we prove consistency of the residual bootstrap in this setting and demonstrate the feasibility of the bootstrap confidence bands at moderate sample sizes in a simulation study.

Suggested Citation

  • Bissantz, Nicolai & Birke, Melanie, 2009. "Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2364-2375, November.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:10:p:2364-2375
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    References listed on IDEAS

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    1. Axel Munk & Nicolai Bissantz & Thorsten Wagner & Gudrun Freitag, 2005. "On difference‐based variance estimation in nonparametric regression when the covariate is high dimensional," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 19-41, February.
    2. Bissantz, Nicolai & Dümbgen, Lutz & Holzmann, Hajo & Munk, Axel, 2007. "Nonparametric confidence bands in deconvolution density estimation," Technical Reports 2007,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. 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.
    4. A. Delaigle & I. Gijbels, 2002. "Estimation of integrated squared density derivatives from a contaminated sample," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 869-886, October.
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    Cited by:

    1. Birke, Melanie & Bissantz, Nicolai, 2012. "Testing for symmetries in multivariate inverse problems," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 236-253.

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