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An Inverse Problem: Recovery of a Distribution Using Wasserstein Barycenter


  • Eustasio DEL BARRIO
  • Jean-Michel LOUBES


We propose to study the inverse problem of estimating a distribution observed through an image by a parametric operator. We construct first estimators for the individual parameters of the operators and study their behaviour, then we invert the approximated operator to obtain and estimator of the distribution. The estimation procedure is given by the minimization of a criterion that compares the alignment of the operator to the Wasserstein barycenter of these warped distributions.

Suggested Citation

  • Eustasio DEL BARRIO & Jean-Michel LOUBES & Bruno PELLETIER, 2017. "An Inverse Problem: Recovery of a Distribution Using Wasserstein Barycenter," Annals of Economics and Statistics, GENES, issue 128, pages 229-259.
  • Handle: RePEc:adr:anecst:y:2017:i:128:p:229-259 DOI: 10.15609/annaeconstat2009.128.0229

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    References listed on IDEAS

    1. Sokullu, Senay, 2016. "Network effects in the German magazine industry," Economics Letters, Elsevier, vol. 143(C), pages 77-79.
    2. Frédérique Fève & Jean-Pierre Florens, 2010. "The practice of non-parametric estimation by solving inverse problems: the example of transformation models," Econometrics Journal, Royal Economic Society, vol. 13(3), pages 1-27, October.
    3. S. Darolles & Y. Fan & J. P. Florens & E. Renault, 2011. "Nonparametric Instrumental Regression," Econometrica, Econometric Society, vol. 79(5), pages 1541-1565, September.
    4. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, number 8355.
    5. Qi Li & Jeffrey Scott Racine, 2006. "Density Estimation, from Nonparametric Econometrics: Theory and Practice," Introductory Chapters,in: Nonparametric Econometrics: Theory and Practice Princeton University Press.
    6. repec:cup:etheor:v:33:y:2017:i:04:p:839-873_00 is not listed on IDEAS
    7. Florens, Jean-Pierre & Sokullu, Senay, 2017. "Nonparametric Estimation Of Semiparametric Transformation Models," Econometric Theory, Cambridge University Press, vol. 33(04), pages 839-873, August.
    8. Joel L. Horowitz, 2011. "Applied Nonparametric Instrumental Variables Estimation," Econometrica, Econometric Society, vol. 79(2), pages 347-394, March.
    9. Senay Sokullu, 2012. "Nonparametric Analysis of Two-Sided Markets," Bristol Economics Discussion Papers 12/628, Department of Economics, University of Bristol, UK.
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    More about this item


    Wasserstein Distance; Inverse Problem; Registration.;

    JEL classification:

    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory


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