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date: 17 October 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

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