Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator
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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Volume (Year): 27 (2011)
Issue (Month): 03 (June)
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Web page: http://journals.cambridge.org/jid_ECT
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Jean‐Pierre Florens & Jan Johannes & Sébastien Van Bellegem, 2012.
"Instrumental regression in partially linear models,"
Royal Economic Society, vol. 15(2), pages 304-324, 06.
- FLORENS, Jean-Pierre & JOHANNES, Jan & VAN BELLEGEM, SÃ©bastien, 2006. "Instrumental regression in partially linear models," CORE Discussion Papers 2006025, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Florens, Jean-Pierre & Johannes, Jan & Van Bellegem, Sébastien, 2009. "Instrumental Regression in Partially Linear Models," TSE Working Papers 10-167, Toulouse School of Economics (TSE).
- 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.
- Jérémie Bigot & Sébastien Van Bellegem, 2009.
"Log-density Deconvolution by Wavelet Thresholding,"
Scandinavian Journal of Statistics,
Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 749-763.
- Bigot, Jérôme & Van Bellegem, Sébastien, 2009. "Log-Density Deconvolution by Wavelet Thresholding," IDEI Working Papers 635, Institut d'Économie Industrielle (IDEI), Toulouse.
- Bigot, Jérôme & Van Bellegem, Sébastien, 2009. "Log-Density Deconvolution by Wavelet Thresholding," TSE Working Papers 09-011, Toulouse School of Economics (TSE).
- 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.
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