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Spectral Method for Deconvolving a Density

  • Carrasco, Marine
  • Florens, Jean-Pierre

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.

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Paper provided by Institut d'Économie Industrielle (IDEI), Toulouse in its series IDEI Working Papers with number 138.

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Date of creation: 2002
Date of revision: 2009
Publication status: Published in Econometric Theory, vol.�27, juin 2011, p.�546-581.
Handle: RePEc:ide:wpaper:1038
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  1. Eric Gautier & Yuichi Kitamura, 2011. "Nonparamatric estimation in random coefficients binary choice models," Working Papers hal-00403939, HAL.
  2. Carrasco, Marine & Florens, Jean-Pierre, 2002. "Simulation-Based Method of Moments and Efficiency," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(4), pages 482-92, October.
  3. Ichimura, Hidehiko & Thompson, T. Scott, 1998. "Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution," Journal of Econometrics, Elsevier, vol. 86(2), pages 269-295, June.
  4. Postel-Vinay & Robin, 2002. "Equilibrium wage dispersion with worker and employer heterogeneity," Working Papers 155908, Institut National de la Recherche Agronomique, France.
  5. Neumann, Michael H., 2007. "Deconvolution from panel data with unknown error distribution," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1955-1968, November.
  6. Fabien Postel-Vinay & Jean-Marc Robin, 2002. "Equilibrium wage dispersion with worker and employer heterogeneity," Sciences Po publications info:hdl:2441/dc0ckec3fcb, Sciences Po.
  7. Horowitz, Joel L & Markatou, Marianthi, 1996. "Semiparametric Estimation of Regression Models for Panel Data," Review of Economic Studies, Wiley Blackwell, vol. 63(1), pages 145-68, January.
  8. P. Groeneboom & G. Jongbloed, 2003. "Density estimation in the uniform deconvolution model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(1), pages 136-157.
  9. Raymond J. Carroll & Peter Hall, 2004. "Low order approximations in deconvolution and regression with errors in variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 31-46.
  10. Geweke, John, 1988. "Antithetic acceleration of Monte Carlo integration in Bayesian inference," Journal of Econometrics, Elsevier, vol. 38(1-2), pages 73-89.
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