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A mollifier approach to the deconvolution of probability densities

Author

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  • Hohage, Thorsten
  • Maréchal, Pierre
  • Simar, Léopold

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Vanhems, Anne

Abstract

We use mollification to regularize the problem of deconvolution of random variables. This regularization method offers a unifying and generalizing framework in order to compare the benefits of various filter-type techniques like deconvolution kernels, Tikhonov, or spectral cutoff methods. In particular, the mollifier approach allows to relax some restrictive assumptions required for the deconvolution kernels, and has better stabilizing properties compared with spectral cutoff or Tikhonov. We show that this approach achieves optimal rates of convergence for both finitely and infinitely smoothing convolution operators under Besov and Sobolev smoothness assumptions on the unknown probability density. The qualification can be arbitrarily high depending on the choice of the mollifier function. We propose an adaptive choice of the regularization parameter using the Lepski ̆ı method, and we provide simulations to compare the finite sample properties of our estimator with respect to the well-known regularization methods.

Suggested Citation

  • Hohage, Thorsten & Maréchal, Pierre & Simar, Léopold & Vanhems, Anne, 2022. "A mollifier approach to the deconvolution of probability densities," LIDAM Reprints ISBA 2022041, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
  • Handle: RePEc:aiz:louvar:2022041
    DOI: https://doi.org/10.1017/S0266466622000457
    Note: In: Econometric Theory, 2022
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    References listed on IDEAS

    as
    1. Carrasco, Marine & Florens, Jean-Pierre, 2011. "A Spectral Method For Deconvolving A Density," Econometric Theory, Cambridge University Press, vol. 27(3), pages 546-581, June.
    2. Carrasco, Marine & Florens, Jean-Pierre & Renault, Eric, 2007. "Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 77, Elsevier.
    3. Bauer, Frank & Lukas, Mark A., 2011. "Comparingparameter choice methods for regularization of ill-posed problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(9), pages 1795-1841.
    4. Aurore Delaigle & Peter Hall, 2016. "Methodology for non-parametric deconvolution when the error distribution is unknown," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 231-252, January.
    5. Kerkyacharian, Gérard & Picard, Dominique, 1993. "Density estimation by kernel and wavelets methods: Optimality of Besov spaces," Statistics & Probability Letters, Elsevier, vol. 18(4), pages 327-336, November.
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