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

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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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