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Log-Density Deconvolution by Wavelet Thresholding

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  • Bigot, Jérôme
  • Van Bellegem, Sébastien

Abstract

This paper proposes a new wavelet-based method for deconvolving a density. The estimator combines the ideas of nonlinear wavelet thresholding with periodised Meyer wavelets and estimation by information projection. It is guaranteed to be in the class of density functions, in particular it is positive everywhere by construction. The asymptotic optimality of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. Finite sample properties is investigated in detail, and show the excellent empirical performance of the estimator, compared with other recently introduced estimators.

Suggested Citation

  • Bigot, Jérôme & Van Bellegem, Sébastien, 2009. "Log-Density Deconvolution by Wavelet Thresholding," TSE Working Papers 09-011, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:22136
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    References listed on IDEAS

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    1. Daniela De Canditiis & Marianna Pensky, 2006. "Simultaneous Wavelet Deconvolution in Periodic Setting," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 293-306, June.
    2. Ja‐Yong Koo, 1999. "Logspline Deconvolution in Besov Space," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(1), pages 73-86, March.
    3. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(3), pages 522-545, June.
    4. JOHANNES, Jan & VAN BELLEGHEM, Sébastien & VANHEMS, Anne, 2007. "A unified approach to solve ill-posed inverse problems in econometrics," LIDAM Discussion Papers CORE 2007083, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Peter Hall & Peihua Qiu, 2005. "Discrete-transform approach to deconvolution problems," Biometrika, Biometrika Trust, vol. 92(1), pages 135-148, March.
    6. Koo, Ja-Yong & Kim, Woo-Chul, 1996. "Wavelet density estimation by approximation of log-densities," Statistics & Probability Letters, Elsevier, vol. 26(3), pages 271-278, February.
    7. 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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    Cited by:

    1. Florens, Jean-Pierre & Schwarz, Maik & Van Bellegem, Sébastien, 2010. "Nonparametric Frontier Estimation from Noisy Data," TSE Working Papers 10-179, Toulouse School of Economics (TSE).
    2. Bak, Kwan-Young & Jhong, Jae-Hwan & Lee, JungJun & Shin, Jae-Kyung & Koo, Ja-Yong, 2021. "Penalized logspline density estimation using total variation penalty," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    3. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(3), pages 522-545, June.

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    Keywords

    deconvolution; wavelet thresholding; adaptive estimation;
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