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Estimation of Convolution In The Model with Noise

Author

Listed:
  • Christophe Chesneau

    (LMNO, Université de Caen Basse-Normandie Département de Mathématiques et UFR de Sciences)

  • Fabienne Comte

    (MAP5, UMR CNRS 8145, Université Paris Descartes, Sorbonne Paris Cité)

  • Gwennaëlle Mabon

    (CREST)

  • Fabien Navarro

    (Université de Nantes, Laboratoire de Mathématiques Jean Leray UFR Sciences et Techniques)

Abstract

We investigate the estimation of the ?-fold convolution of the density of an unob- served variable X from n i.i.d. observations of the convolution model Y = X + ?. We first assume that the density of the noise ? is known and define nonadaptive estimators, for which we provide bounds for the mean integrated squared error (MISE). In particular, under some smoothness assumptions on the densities of X and ?, we prove that the parametric rate of con-vergence 1/n can be attained. Then we construct an adaptive estimator using a penalization approach having similar performances to the nonadaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.

Suggested Citation

  • Christophe Chesneau & Fabienne Comte & Gwennaëlle Mabon & Fabien Navarro, 2014. "Estimation of Convolution In The Model with Noise," Working Papers 2014-39, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2014-39
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    References listed on IDEAS

    as
    1. Stéphane Bonhomme & Jean-Marc Robin, 2010. "Generalized Non-Parametric Deconvolution with an Application to Earnings Dynamics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 77(2), pages 491-533.
    2. Kappus, Johanna, 2014. "Adaptive nonparametric estimation for Lévy processes observed at low frequency," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 730-758.
    3. Johanna Kappus & Gwennaelle Mabon, 2013. "Adaptive Density Estimation in Deconvolution Problems with Unknown Error Distribution," Working Papers 2013-31, Center for Research in Economics and Statistics.
    4. Mugdadi, A. R. & Ahmad, Ibrahim A., 2004. "A bandwidth selection for kernel density estimation of functions of random variables," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 49-62, August.
    5. B. W. Silverman, 1982. "Kernel Density Estimation Using the Fast Fourier Transform," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 31(1), pages 93-99, March.
    6. F. Comte & C. Lacour, 2011. "Data‐driven density estimation in the presence of additive noise with unknown distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 601-627, September.
    7. Neumann, Michael H., 2007. "Deconvolution from panel data with unknown error distribution," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1955-1968, November.
    8. A. Ahmad, Ibrahim & Fan, Yanqin, 2001. "Optimal bandwidths for kernel density estimators of functions of observations," Statistics & Probability Letters, Elsevier, vol. 51(3), pages 245-251, February.
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