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A mollifier approach to the deconvolution of probability densities

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  • Marechal, Pierre
  • Simar, Leopold
  • 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 cut- off methods. In particular, the mollifier approach allows to relax some restrictive assumptions required for the deconvolution kernels, and has better stabilizing properties compared to spectral cutoff or Tikhonov. We show that this approach achieves optimal rates of convergence both for 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 Lepskii method and we provide simulations to compare the finite sample properties of our estimator with respect to the well-known regularization methods.
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Suggested Citation

  • Marechal, Pierre & Simar, Leopold & Vanhems, Anne, 2018. "A mollifier approach to the deconvolution of probability densities," LIDAM Discussion Papers ISBA 2018028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
  • Handle: RePEc:aiz:louvad:2018028
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    References listed on IDEAS

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    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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