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A SPECTRAL METHOD FOR DECONVOLVING A DENSITY

Published online by Cambridge University Press:  11 October 2010

Marine Carrasco  and
Jean-Pierre Florens
Marine Carrasco*
Affiliation:
University of Montreal
Jean-Pierre Florens
Affiliation:
Toulouse School of Economics
*
*Address correspondence to Marine Carrasco, Universite de Montreal, Departement de Sciences Economiques, CP 6128, succ Centre Ville, Montreal, QC H3C3J7, Canada; e-mail: [email protected].
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Abstract

We propose a new estimator for the density of a random variable observed with an additive measurement error. This estimator is based on the spectral decomposition of the convolution operator, which is compact for an appropriate choice of reference spaces. The density is approximated by a sequence of orthonormal eigenfunctions of the convolution operator. The resulting estimator is shown to be consistent and asymptotically normal. While most estimation methods assume that the characteristic function (CF) of the error does not vanish, we relax this assumption and allow for isolated zeros. For instance, the CF of the uniform and symmetrically truncated normal distributions have isolated zeros. We show that, in the presence of zeros, the density is identified even though the convolution operator is not one-to-one. We propose two consistent estimators of the density. We apply our method to the estimation of the measurement error density of hourly income collected from survey data.

Type
ARTICLES
Information
Econometric Theory , Volume 27 , Issue 3: SPECIAL ISSUE ON INVERSE PROBLEMS IN ECONOMETRICS , June 2011 , pp. 546 - 581
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Billingsley, P. (1995) Probability and Measure. Wiley.Google Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica 75, 1613–1669.CrossRefGoogle Scholar
Carrasco, M. & Florens, J.P. (2002) Simulation based method of moments and efficiency. Journal of Business & Economic Statistics 20(4), 482–492.Google 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.J. & Leamer, E.E. (eds.), The Handbook of Econometrics, vol. 6, pp. 5633–5751. Elsevier.CrossRefGoogle Scholar
Carroll, R. & Hall, P. (1988) Optimal rates of convergence for deconvolving a density. Journal of American Statistical Association 83(404), 1184–1186.CrossRefGoogle Scholar
Carroll, R. & Hall, P. (2004) Low order approximations in deconvolution and regression with errors in variables. Journal of the Royal Statistical Society Series B 66, 31–46.CrossRefGoogle Scholar
Carroll, R., Van Rooij, A., & Ruymgaart, F. (1991) Theoretical aspects of iii-posed problems in statistics. Acta Applicandae Mathematicae 24, 113–140.CrossRefGoogle Scholar
Chernozhukov, V., Gagliardini, P., & Scaillet, O. (2008) Nonparametric Instrumental Variable, Estimation of Quantile Structural Effects. Working paper, Swiss Finance Institute.Google Scholar
Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer Verlag.CrossRefGoogle Scholar
Devroye, L. (1989) Consistent deconvolution in density estimation. Canadian Journal of Statistics 17, 235–239.Google Scholar
Donoho, D. (1995) Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied and Computational Harmonic Analysis 2, 101–126.CrossRefGoogle Scholar
Dunford, N. & Schwartz, J. (1963) Linear Operators, Part II, Spectral Theory. Wiley.Google Scholar
Efromovich (1997) Density estimation for the case of supersmooth measurement error. Journal of the American Statistical Association 92(438), 526–535.CrossRefGoogle Scholar
Engl, H.W., Hanke, M., & Neubauer, A. (1996) Regularization of Inverse Problems. Kluwer Academic.CrossRefGoogle Scholar
Fan, J. (1991a) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 1257–1272.CrossRefGoogle Scholar
Fan, J. (1991b) Asymptotic normality for deconvolution kernel density estimators. Sankhya 53, 97–110.Google Scholar
Fan, J. (1993) Adaptively local one-dimentional subproblems with application to a deconvolution problem. Annals of Statistics 21, 600–610.CrossRefGoogle Scholar
Gautier, E. & Kitamura, Y. (2008) Nonparametric Estimation in Random Coefficients Binary Choice Models. Manuscript, Yale University.CrossRefGoogle Scholar
Geweke, J. (1988) Antithetic acceleration of Monte Carlo integration in Bayesian inference. Journal of Econometrics 38, 73–90.CrossRefGoogle Scholar
Gourieroux, C. & Monfort, A. (1996) Simulation-Based Econometric Methods. Oxford University Press.Google Scholar
Groeneboom, P. & Jongbloed, G. (2003) Density estimation in the uniform deconvolution model. Statistica Neerlandica 57, 136–157.Google Scholar
Hall, P. & Horowitz, J. (2005) Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 33, 2904–2929.CrossRefGoogle Scholar
Hall, P. & Meister, A. (2007) A ridge-parameter approach to deconvolution. Annals of Statistics 35, 1535–1558.CrossRefGoogle Scholar
Hall, P., Ruymgaart, F., van Gaans, O., & van Rooij, A. (2001) Inverting noisy integral equations using wavelet expansions: A class of irregular convolutions. In de Gunst, M.C.M., Klassen, C.A.J., & van der Vaart, A.W. (eds.), Festschrift in Honor of Willem van Zwet. IMS Lectures Notes Monograph Series, vol. 36, pp. 533–546. Institute of Mathematical Statistics.Google Scholar
Horowitz, J. & Markatou, M. (1996) Semiparametric estimation of regression models for panel data. Review of Economic Studies 63, 145–168.CrossRefGoogle Scholar
Hu, Y. & Ridder, G. (2007) Estimation of Nonlinear Models with Mismeasured Regressors Using Marginal Information. Mimeo, University of Southern California.Google Scholar
Ichimura, H. & Thompson, T.S. (1998) Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution. Journal of Econometrics 86, 269–295.CrossRefGoogle Scholar
Johannes, J. (2009) Deconvolution with unknown error distribution. Annals of Statistics 39, 2301–2323.Google Scholar
Johnstone, I., Kerkyacharian, G., Picard, D., & Raimondo, M. (2004) Wavelet decomposition in a periodic setting. Journal of the Royal Statistical Society Series B, 66, 547–573.CrossRefGoogle Scholar
Johnstone, I. & Raimondo, M. (2004) Periodic boxcar deconvolution and diophantine approximation. Annals of Statistics 32, 1781–1804.CrossRefGoogle Scholar
Kress, R. (1999) Linear Integral Equations. Springer.Google Scholar
Lukacs, E. (1970) Characteristic Functions. Statistical Monographs and Courses. Griffin.Google Scholar
Meister, A. (2007) Deconvolving compactly supported densities. Mathematical Methods of Statistics 16, 63–76.CrossRefGoogle Scholar
Meister, A. (2008) Deconvolving from Fourier-oscillating error densities under decay and smoothness restrictions. Inverse Problems 24, 1–14.CrossRefGoogle Scholar
Nashed, M.Z. & Wahba, G. (1974) Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations. SIAM Journal on Mathematical Analysis 5, 974–987.CrossRefGoogle Scholar
Neubauer, A. (1987) Finite-dimensional approaximation of constrained Tikhonov-regularized solutions of iii-posed linear operator equations. Mathematics of Computation 48, 565–583.Google Scholar
Neumann, M. (2007) Deconvolution from panel data with unknown error distribution. Journal of Multivariate Analysis 98, 1955–1968.Google Scholar
Pensky, M. & Vidakovic, B. (1999) Adaptive wavelet estimator for nonparametric dendity deconvolution. Annals of Statistics 27, 2033–2053.CrossRefGoogle Scholar
Postel-Vinay, F. & Robin, J.-M. (2002) Equilibrium wage dispersion with worker and employer heterogeneity. Econometrica 70, 2295–2350.CrossRefGoogle Scholar
Serfling, R. (1980) Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Stefanski, L. & Carroll, R. (1990) Deconvoluting kernel density estimators. Statistics 2, 169–184.CrossRefGoogle Scholar
Tikhonov, A. & Arsenin, V. (1977) Solutions of Ill-posed Problems. Winston & Sons.Google Scholar
Ushakov, N. (1999) Selected Topics in Characteristic Functions. VSP.CrossRefGoogle Scholar
Van Rooij, A.C.M. & Ruymgaart, F.H. (1991) Regularized deconvolution on the circle and the sphere. In Roussas, G. (ed.), Nonparametric Functional Estimation and Related Topics, pp. 679–690. Kluwer Academic.Google Scholar
Van Rooij, A.C.M. & Ruymgaart, F.H. (1999) On inverse estimation. In Ghosh, S. (ed.), Asymptotics, Nonparametrics and Time Series, pp. 579–613. Dekker.Google Scholar
Walter, G. (1981) Orthogonal series estimators of the prior distribution. Sankhia: The Indian Journal of Statistics Series A 43, 228–245.Google Scholar
Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing, Chapman and Hall.CrossRefGoogle Scholar