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Convergence Rates for III-Posed Inverse Problems with an Unknown Operator

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  • Johannes, Jan
  • Van Bellegem, Sébastien
  • Vanhems, Anne

Abstract

This paper studies the estimation of a nonparametric function ' from the inverse problem r = T' given estimates of the function r and of the linear transform T. The rate of convergence of the estimator is derived under two assumptions expressed in a Hilbert scale. The approach provides a unified framework that allows to compare various sets of structural assumptions used in the econometrics literature. General upper bounds are derived for the risk of the estimator of the structural function ' as well as of its derivatives. It is shown that the bounds cover and extend known results given in the literature. Particularly, they imply new results in two applications. The first application is the blind nonparametric deconvolution on the real line, and the second application is the estimation of the derivatives of the nonparametric instrumental regression function via an iterative Tikhonov regularization scheme.

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Bibliographic Info

Paper provided by Toulouse School of Economics (TSE) in its series TSE Working Papers with number 09-030.

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Date of creation: 03 Apr 2009
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Publication status: Published in Econometric Theory, vol.�27, n°3, juin 2011, p.�522-545.
Handle: RePEc:tse:wpaper:22144

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Keywords: inverse problem; Hibert scale; blind deconvolution;

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References

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  1. Florens, Jean-Pierre & Johannes, Jan & Van Bellegem, Sébastien, 2009. "Instrumental Regression in Partially Linear Models," IDEI Working Papers 613, Institut d'Économie Industrielle (IDEI), Toulouse.
  2. Bigot, Jérôme & Van Bellegem, Sébastien, 2009. "Log-Density Deconvolution by Wavelet Thresholding," TSE Working Papers 09-011, Toulouse School of Economics (TSE).
  3. Carrasco, Marine & Florens, Jean-Pierre, 2002. "Spectral Method for Deconvolving a Density," IDEI Working Papers 138, Institut d'Économie Industrielle (IDEI), Toulouse, revised 2009.
  4. Fabien Postel-Vinay & Jean-Marc Robin, 2002. "Equilibrium Wage Dispersion with Worker and Employer Heterogeneity," Econometrica, Econometric Society, vol. 70(6), pages 2295-2350, November.
  5. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, 09.
  6. Richard Blundell & Xiaohong Chen & Dennis Kristensen, 2003. "Nonparametric IV estimation of shape-invariant Engel curves," CeMMAP working papers CWP15/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  7. Richard Blundell & Xiaohong Chen & Dennis Kristensen, 2007. "Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves," Econometrica, Econometric Society, vol. 75(6), pages 1613-1669, November.
  8. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  9. 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.
  10. Stéphane Bonhomme & Jean-Marc Robin, 2008. "Generalized nonparametric deconvolution with an application to earnings dynamics," CeMMAP working papers CWP03/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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Cited by:
  1. Schwarz, Maik & Van Bellegem, Sébastien, 2009. "Consistent Density Deconvolution under Partially Known Error Distribution," TSE Working Papers 09-097, Toulouse School of Economics (TSE).
  2. Anna Simoni & Jean-Pierre Florens, 2013. "Regularizing Priors for Linear Inverse Problems," THEMA Working Papers 2013-32, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  3. Jean-Pierre Florens & Anna Simoni, 2013. "Regularizing Priors for Linear Inverse Problems," Working Papers hal-00873180, HAL.

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