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)
|Contact details of provider:|| Postal: Cambridge University Press, UPH, Shaftesbury Road, Cambridge CB2 8BS UK|
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.:
- FLORENS, Jean-Pierre & JOHANNES, Jan & VAN BELLEGEM, Sébastien, 2006.
"Instrumental regression in partially linear models,"
CORE Discussion Papers
2006025, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Jean‐Pierre Florens & Jan Johannes & Sébastien Van Bellegem, 2012. "Instrumental regression in partially linear models," Econometrics Journal, Royal Economic Society, vol. 15(2), pages 304-324, 06.
- 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.
- 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.
When requesting a correction, please mention this item's handle: RePEc:cup:etheor:v:27:y:2011:i:03:p:522-545_00. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Keith Waters)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.