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CONVERGENCE RATES FOR ILL-POSED INVERSE PROBLEMS WITH AN UNKNOWN OPERATOR

Published online by Cambridge University Press:  11 October 2010

Jan Johannes ,
Sébastien Van Bellegem  and
Anne Vanhems
Jan Johannes
Affiliation:
Université catholique de Louvain
Sébastien Van Bellegem*
Affiliation:
Toulouse School of Economics and CORE
Anne Vanhems
Affiliation:
Toulouse School of Economics and Toulouse Business School
*
*Address correspondence to Sebastien Van Bellegem, 21 Allée de Brienne, 31000 Toulouse, France; e-mail: [email protected].
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Abstract

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.

Type
ARTICLES
Information
Econometric Theory , Volume 27 , Issue 3: SPECIAL ISSUE ON INVERSE PROBLEMS IN ECONOMETRICS , June 2011 , pp. 522 - 545
Copyright
Copyright © Cambridge University Press 2011

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References

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