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Nonparametric confidence bands in deconvolution density estimation

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  • Bissantz, Nicolai
  • Dümbgen, Lutz
  • Holzmann, Hajo
  • Munk, Axel
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    Abstract

    Uniform confidence bands for densities f via nonparametric kernel estimates were first constructed by Bickel and Rosenblatt [Ann. Statist. 1, 1071.1095]. In this paper this is extended to confidence bands in the deconvolution problem g = f for an ordinary smooth error density . Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (LÉ-distance) between a deconvolution kernel estimator . f and f. Further consistency of the simple nonparametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data-driven bandwidth selector based on heuristic arguments, which aims --

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    Bibliographic Info

    Paper provided by Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen in its series Technical Reports with number 2007,03.

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    Date of creation: 2007
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    Handle: RePEc:zbw:sfb475:200703

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    References

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    1. Delaigle, A. & Gijbels, I., 2001. "Bootstrap Bandwidth Selection in Kernel Density Estimation from a Contaminated Sample," Papers 0116, Catholique de Louvain - Institut de statistique.
    2. Härdle, Wolfgang, 1989. "Asymptotic maximal deviation of M-smoothers," Journal of Multivariate Analysis, Elsevier, vol. 29(2), pages 163-179, May.
    3. Delaigle, A. & Gijbels, I., 2004. "Practical bandwidth selection in deconvolution kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 249-267, March.
    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.
    5. Politis, Dimitris N. & Romano, Joseph P., 1999. "Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 1-25, January.
    6. Bert Van Es & Hae-Won Uh, 2005. "Asymptotic Normality of Kernel-Type Deconvolution Estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 32(3), pages 467-483.
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    Cited by:
    1. Bissantz, Nicolai & Holzmann, Hajo & Proksch, Katharina, 2014. "Confidence regions for images observed under the Radon transform," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 86-107.
    2. Shih-Kang Chao & Katharina Proksch & Holger Dette & Wolfgang Härdle, 2014. "Confidence Corridors for Multivariate Generalized Quantile Regression," SFB 649 Discussion Papers SFB649DP2014-028, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Anti-concentration and honest, adaptive confidence bands," CeMMAP working papers CWP69/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Bissantz, Nicolai & Birke, Melanie, 2008. "Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators," Technical Reports 2008,17, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    5. Wang, Xiao-Feng & Fan, Zhaozhi & Wang, Bin, 2010. "Estimating smooth distribution function in the presence of heteroscedastic measurement errors," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 25-36, January.
    6. 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.
    7. Birke, Melanie & Bissantz, Nicolai & Holzmann, Hajo, 2008. "Confidence bands for inverse regression models with application to gel electrophoresis," Technical Reports 2008,16, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.

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