Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator
This paper studies the estimation of a nonparametric function ϕ from the inverse problem r = Tϕ given estimates of the function r and of the linear transform T . We show that rates of convergence of the estimator are driven by two types of assumptions expressed in a single Hilbert scale. The two assumptions quantify the prior regularity of ϕ and the prior link existing between T and the Hilbert scale. The approach provides a unified framework that allows us to compare various sets of structural assumptions found in the econometric literature. Moreover, general upper bounds are also derived for the risk of the estimator of the structural function ϕ as well as that of its derivatives. It is shown that the bounds cover and extend known results given in the literature. Two important applications are also studied. The first is the blind nonparametric deconvolution on the real line, and the second is the estimation of the derivatives of the nonparametric instrumental regression function via an iterative Tikhonov regularization scheme.
Volume (Year): 27 (2011)
Issue (Month): 03 (June)
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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.:
- 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.
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- 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).
- Jérémie Bigot & Sébastien Van Bellegem, 2009.
"Log-density Deconvolution by Wavelet Thresholding,"
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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," TSE Working Papers 09-011, Toulouse School of Economics (TSE).
- Bigot, Jérôme & Van Bellegem, Sébastien, 2009. "Log-Density Deconvolution by Wavelet Thresholding," IDEI Working Papers 635, Institut d'Économie Industrielle (IDEI), Toulouse.
- 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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