Show Summary Details

Page of

PRINTED FROM OXFORD HANDBOOKS ONLINE (www.oxfordhandbooks.com). © Oxford University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a title in Oxford Handbooks Online for personal use (for details see Privacy Policy and Legal Notice).

date: 16 July 2019

Abstract and Keywords

This chapter studies the estimation of φ in linear inverse problems Tφ = r, where r is only observed with error and T may be given or estimated. The unknown element φ belongs to a Hilbert space E. Four examples are relevant for econometrics: the density estimation, the deconvolution problem, the linear regression with an infinite number of possibly endogenous explanatory variables, and the nonparametric instrumental variables estimation. In the first two cases T is given, whereas it is estimated in the two other cases, respectively at a parametric or nonparametric rate. This chapter will recall the main results on these models: concepts of degree of ill-posedness, regularity of φ, regularized estimation, and the rates of convergence usually obtained. The main contributions are, moreover, related to the asymptotic normality of the regularized solution φ obtained with a regularization parameter α. If α → 0, we particularly consider the asymptotic normality of inner products <φ, ϕ>, where ϕ is an element of E. These results can be used to construct (asymptotic) tests on φ.

Keywords: deconvolution, functional linear regression, nonparametric instrumental regression, ill-posed inverse problem, Tikhonov regularization, Hilbert scales, asymptotic normality

Access to the complete content on Oxford Handbooks Online requires a subscription or purchase. Public users are able to search the site and view the abstracts and keywords for each book and chapter without a subscription.

Please subscribe or login to access full text content.

If you have purchased a print title that contains an access token, please see the token for information about how to register your code.

For questions on access or troubleshooting, please check our FAQs, and if you can''t find the answer there, please contact us.