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Asymptotic Normality of Kernel‐Type Deconvolution Estimators

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  • BERT VAN ES
  • HAE‐WON UH

Abstract

. We derive asymptotic normality of kernel‐type deconvolution estimators of the density, the distribution function at a fixed point, and of the probability of an interval. We consider so‐called super smooth deconvolution problems where the characteristic function of the known distribution decreases exponentially, but faster than that of the Cauchy distribution. It turns out that the limit behaviour of the pointwise estimators of the density and distribution function is relatively straightforward, while the asymptotic behaviour of the estimator of the probability of an interval depends in a complicated way on the sequence of bandwidths.

Suggested Citation

  • 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, September.
    • Handle: RePEc:bla:scjsta:v:32:y:2005:i:3:p:467-483
    DOI: 10.1111/j.1467-9469.2005.00443.x
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    Cited by:

    1. Adusumilli, Karun & Kurisu, Daisuke & Otsu, Taisuke & Whang, Yoon-Jae, 2020. "Inference on distribution functions under measurement error," Journal of Econometrics, Elsevier, vol. 215(1), pages 131-164.
    2. Peter Hall & Tapabrata Maiti, 2008. "Non‐parametric inference for clustered binary and count data when only summary information is available," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(4), pages 725-738, September.
    3. Dong, Hao & Otsu, Taisuke & Taylor, Luke, 2021. "Average Derivative Estimation Under Measurement Error," Econometric Theory, Cambridge University Press, vol. 37(5), pages 1004-1033, October.
    4. Yousri Slaoui, 2021. "Data-driven Deconvolution Recursive Kernel Density Estimators Defined by Stochastic Approximation Method," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 312-352, February.
    5. 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.
    6. Otsu, Taisuke & Taylor, Luke, 2021. "Specification Testing For Errors-In-Variables Models," Econometric Theory, Cambridge University Press, vol. 37(4), pages 747-768, August.
    7. van Es, Bert & Gugushvili, Shota, 2008. "Weak convergence of the supremum distance for supersmooth kernel deconvolution," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2932-2938, December.
    8. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    9. Dong, Hao & Taylor, Luke, 2022. "Nonparametric Significance Testing In Measurement Error Models," Econometric Theory, Cambridge University Press, vol. 38(3), pages 454-496, June.
    10. Martin L. Hazelton & Berwin A. Turlach, 2010. "Semiparametric Density Deconvolution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 91-108, March.
    11. Adusumilli, Karun & Kurisu, Daisies & Otsu, Taisuke & Whang, Yoon-Jae, 2020. "Inference on distribution functions under measurement error," LSE Research Online Documents on Economics 102692, London School of Economics and Political Science, LSE Library.
    12. Mynbaev, Kairat, 2011. "Distributions escaping to infinity and the limiting power of the Cliff-Ord test for autocorrelation," MPRA Paper 44402, University Library of Munich, Germany, revised 18 Sep 2012.
    13. Ali Al-Sharadqah & Majid Mojirsheibani & William Pouliot, 2020. "On the performance of weighted bootstrapped kernel deconvolution density estimators," Statistical Papers, Springer, vol. 61(4), pages 1773-1798, August.
    14. Yang Zu, 2015. "A Note on the Asymptotic Normality of the Kernel Deconvolution Density Estimator with Logarithmic Chi-Square Noise," Econometrics, MDPI, vol. 3(3), pages 1-16, July.
    15. Mynbaev, Kairat & Martins-Filho, Carlos, 2015. "Consistency and asymptotic normality for a nonparametric prediction under measurement errors," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 166-188.
    16. Jakob Söhl & Mathias Trabs, 2012. "We estimate linear functionals in the classical deconvolution problem by kernel estimators," SFB 649 Discussion Papers SFB649DP2012-046, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    17. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.

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