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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

REFERENCES

Bigot, J. & Van Bellegem, S. (2009) Log-density deconvolution by wavelet thresholding. Scandinavian Journal of Statistics 36, 749–763.CrossRefGoogle Scholar
Bissantz, N., Hohage, T., Munk, A., & Ruymgaart, F.H. (2007) Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM Journal on Numerical Analysis 45(6), 2610–2636.CrossRefGoogle Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica 75, 1613–1669.CrossRefGoogle Scholar
Bonhomme, S. & Robin, J.-M. (2010) Generalized non-parametric deconvolution with an application to earnings dynamics. Review of Economic Studies 77, 491–533.CrossRefGoogle Scholar
Carrasco, M. & Florens, J.-P. (2011) A spectral method for deconvolving a density. Econometric Theory 27 (this issue).CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics: Estimation based on spectral decomposition and regularization. In Heckman, J. & Leamer, E. (eds.), Handbook of Econometrics, vol. 6B, pp. 5633–5746. Elsevier.CrossRefGoogle Scholar
Cavalier, L. & Hengartner, N. (2005) Adaptive estimation for inverse problems with noisy operators. Inverse Problems 21, 1345–1361.Google Scholar
Chen, X. & Reiss, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27 (this issue).Google Scholar
Darolles, S., Florens, J.-P., & Renault, E. (2002) Nonparametric Instrumental Regression. Working paper 228, IDEI, Université de Toulouse I.Google Scholar
Egger, H. (2005) Accelerated Newton-Landweber Iterations for Regularizing Nonlinear Inverse Problems. SFB-Report 2005-3, Austrian Academy of Sciences, Linz.Google Scholar
Fan, J. (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 1257–1272.CrossRefGoogle Scholar
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2005) Instrumental Regression in Partially Linear Models. Discussion paper 0537, Université catholique de Louvain.CrossRefGoogle Scholar
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2011) Identification and estimation by penalization in nonparametric instrumental regression. Econometric Theory 27 (this issue).CrossRefGoogle Scholar
Hall, P. & Horowitz, J. L. (2005) Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 33, 2904–2929.CrossRefGoogle Scholar
Halmos, P.R. (1963) What does the spectral theorem say? American Mathematics Monthly 70, 241–247.CrossRefGoogle Scholar
Hohage, T. (2000) Regularisation of exponentially ill-posed problems. Numerical Functional Analysis and Optimization 21, 439–464.Google Scholar
Horowitz, J.L. (1998) Semiparametric Methods in Econometrics. Springer.CrossRefGoogle Scholar
Johannes, J. (2009) Deconvolution with unknown error density. Annals of Statistics 37, 2301–2323.CrossRefGoogle Scholar
Johannes, J., Van Bellegem, S., & Vanhems, A. (2010) Projection Estimation in Nonparametric Instrumental Regression. Working paper, CORE, Université catholique de Louvain.Google Scholar
Kawata, T. (1972) Fourier Analysis in Probability Theory. Academic Press.Google Scholar
Krein, S. & Petunin, Y.I. (1966) Scales of Banach spaces. Russian Mathematical Surveys 21, 85–169.CrossRefGoogle Scholar
Mair, B.A. (1994) Tikhonov regularization for finitely and infinitely smoothing operators. SIAM Journal on Mathematical Analysis 25, 135–147.CrossRefGoogle Scholar
Mair, B.A. & Ruymgaart, F.H. (1996) Statistical inverse estimation in Hilbert scales. SIAM Journal on Applied Mathematics 56(5), 1424–1444.Google Scholar
Nair, M., Pereverzev, S.V., & Tautenhahn, U. (2005) Regularization in Hilbert scales under general smoothing conditions. Inverse Problems 21, 1851–1869.CrossRefGoogle Scholar
Natterer, F. (1984) Error bounds for Tikhonov regularization in Hilbert scales. Applicable Analysis, 18, 29–37.CrossRefGoogle Scholar
Neubauer, A. (1988) When do Sobolev spaces form a Hilbert scale? Proceedings of the American Mathematical Society 103(2), 557–562.CrossRefGoogle Scholar
Neumann, M.H. (1997) On the effect of estimating the error density in nonparametric deconvolution. Journal of Nonparametric Statistics 7, 307–330.Google Scholar
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578.CrossRefGoogle Scholar
Olver, F. (1974) Asymptotics and Special Functions. Academic Press.Google Scholar
Petrov, V.V. (1995) Limit Theorems of Probability Theory. Sequences of Independent Random Variables., Oxford Studies in Probability 4th ed. Clarendon Press.Google Scholar
Postel-Vinay, F. & Robin, J.-M. (2002) Equilibrium wage dispersion with worker and employer heterogeneity. Econometrica 70, 2295–2350.CrossRefGoogle Scholar
Schwarz, M. & Van Bellegem, S. (2010) Consistent density deconvolution under partially known error distribution. Statistics & Probability Letters 80, 236–241.Google Scholar
Tautenhahn, U. (1996) Error estimates for regularization methods in Hilbert scales. SIAM Journal on Numerical Analysis 33(6), 2120–2130.CrossRefGoogle Scholar