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Convergence rates for ill-posed inverse problems with an unknown operator

Onglets principaux

Johannes, Jan [UCL] Van Bellegem, Sébastien [UCL] Vanhems, Anne [Toulouse Business School]

This paper studies the estimation of a nonparametric function \varphi from the inverse problem r = T \varphi 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 \varphi 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 \varphi 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 de document Article de périodique (Journal article) – Article de recherche
Type d'accès Accès interdit
Année de publication 2011
Langue Anglais
Information sur le périodique "Econometric Theory" - Vol. 27, no. 3, p. 522-545 (2011)
Peer reviewed oui
Editeur Cambridge University Press ((United Kingdom) Cambridge)
issn 0266-4666
e-issn 1469-4360
Statut de la publication Publié
Affiliations UCL - SSH/LIDAM/CORE - Center for operations research and econometrics
Toulouse Business School -
UCL - EUEN/CORE - Center for operations research and econometrics
Liens
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Référence bibliographique Johannes, Jan ; Van Bellegem, Sébastien ; Vanhems, Anne. Convergence rates for ill-posed inverse problems with an unknown operator. In: Econometric Theory, Vol. 27, no. 3, p. 522-545 (2011)
Permalien http://hdl.handle.net/2078.1/87912