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We estimate linear functionals in the classical deconvolution problem by kernel estimators

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  • Jakob Söhl
  • Mathias Trabs

Abstract

We obtain a uniform central limit theorem with square root n rate on the assumption that the smoothness of the functionals is larger than the ill-posedness of the problem, which is given by the polynomial decay rate of the characteristic function of the error. The limit distribution is a generalized Brownian bridge with a covariance structure that depends on the characteristic function of the error and on the functionals. The proposed estimators are optimal in the sense of semiparametric efficiency. The class of linear functionals is wide enough to incorporate the estimation of distribution functions. The proofs are based on smoothed empirical processes and mapping properties of the deconvolution operator.

Suggested Citation

  • 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.
  • Handle: RePEc:hum:wpaper:sfb649dp2012-046
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2012-046.pdf
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    References listed on IDEAS

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    1. 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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    Keywords

    Deconvolution; Donsker theorem; Efficiency; Distribution function; Smoothed empirical processes; Fourier multiplier;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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