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A mollifier approach to the deconvolution of probability densities

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  • Maréchal, Pierre
  • Simar, Léopold
  • Vanhems, Anne

Abstract

In this paper, we use a mollifier approach to regularize the deconvolution, which has been used in research fields like medical imaging, tomography, astrophysics but, to the best of our knowledge, never in statistics or econometrics. We show that the analysis of this new regularization method offers a unifying and generalizing frame in order to compare the benefits of various different filter-type techniques like deconvolution kernels, Tikhonov or spectral cut-off method. In particular, the mollifier approach allows to relax some restrictive assumptions required for the deconvolution problem, and has better stabilizing properties compared to spectral cutoff and Tikhonov. We prove the asymptotic convergence of our estimator and provide simulations analysis to compare the finite sample properties of our estimator with respect to the well-known methods.

Suggested Citation

  • Maréchal, Pierre & Simar, Léopold & Vanhems, Anne, 2018. "A mollifier approach to the deconvolution of probability densities," TSE Working Papers 18-965, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:33097
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    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
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