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

Jan Johannes, Sebastien Van Bellegem () and Anne Vanhems ()

No 09-030, TSE Working Papers from Toulouse School of Economics (TSE)

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.

Keywords: inverse problem; Hibert scale; blind deconvolution (search for similar items in EconPapers)
JEL-codes: C14 C30 (search for similar items in EconPapers)
Date: 2009-04
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Citations: View citations in EconPapers (1)

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Related works:
Journal Article: CONVERGENCE RATES FOR ILL-POSED INVERSE PROBLEMS WITH AN UNKNOWN OPERATOR (2011)
Working Paper: Convergence rates for ill-posed inverse problems with an unknown operator (2011)
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Persistent link: https://EconPapers.repec.org/RePEc:tse:wpaper:22144

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