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A Spectral Method For Deconvolving A Density

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  • Carrasco, Marine
  • Florens, Jean-Pierre

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.

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  • Carrasco, Marine & Florens, Jean-Pierre, 2011. "A Spectral Method For Deconvolving A Density," Econometric Theory, Cambridge University Press, vol. 27(3), pages 546-581, June.
  • Handle: RePEc:cup:etheor:v:27:y:2011:i:03:p:546-581_00
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    2. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2020. "Robust frontier estimation from noisy data: A Tikhonov regularization approach," Econometrics and Statistics, Elsevier, vol. 14(C), pages 1-23.
    3. Manuel Arellano & Richard Blundell & Stéphane Bonhomme, 2017. "Earnings and Consumption Dynamics: A Nonlinear Panel Data Framework," Econometrica, Econometric Society, vol. 85, pages 693-734, May.
    4. Evdokimov, Kirill & White, Halbert, 2012. "Some Extensions Of A Lemma Of Kotlarski," Econometric Theory, Cambridge University Press, vol. 28(4), pages 925-932, August.
    5. Babii, Andrii, 2020. "Honest Confidence Sets In Nonparametric Iv Regression And Other Ill-Posed Models," Econometric Theory, Cambridge University Press, vol. 36(4), pages 658-706, August.
    6. Jarociński, Marek & Marcet, Albert, 2019. "Priors about observables in vector autoregressions," Journal of Econometrics, Elsevier, vol. 209(2), pages 238-255.
    7. S. Darolles & Y. Fan & J. P. Florens & E. Renault, 2011. "Nonparametric Instrumental Regression," Econometrica, Econometric Society, vol. 79(5), pages 1541-1565, September.
    8. Schennach, Susanne M., 2019. "Convolution without independence," Journal of Econometrics, Elsevier, vol. 211(1), pages 308-318.
    9. Gagliardini, Patrick & Scaillet, Olivier, 2012. "Tikhonov regularization for nonparametric instrumental variable estimators," Journal of Econometrics, Elsevier, vol. 167(1), pages 61-75.
    10. Christophe Gaillac & Eric Gautier, 2021. "Nonparametric classes for identification in random coefficients models when regressors have limited variation," Working Papers hal-03231392, HAL.
    11. Manuel Arellano & Stephane Bonhomme, 2019. "Recovering Latent Variables by Matching," Papers 1912.13081, arXiv.org.
    12. Irene Botosaru & Chris Muris & Senay Sokullu, 2022. "Time-Varying Linear Transformation Models with Fixed Effects and Endogeneity for Short Panels," Department of Economics Working Papers 2022-01, McMaster University.
    13. Otsu, Taisuke & Taylor, Luke, 2021. "Specification Testing For Errors-In-Variables Models," Econometric Theory, Cambridge University Press, vol. 37(4), pages 747-768, August.
    14. Marechal, Pierre & Simar, Leopold & Vanhems, Anne, 2018. "A mollifier approach to the deconvolution of probability densities," LIDAM Discussion Papers ISBA 2018028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Frédérique Fève & Jean-Pierre Florens, 2010. "The practice of non-parametric estimation by solving inverse problems: the example of transformation models," Econometrics Journal, Royal Economic Society, vol. 13(3), pages 1-27, October.
    16. Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Robust inference in deconvolution," Quantitative Economics, Econometric Society, vol. 12(1), pages 109-142, January.
    17. Centorrino, Samuele & Florens, Jean-Pierre, 2021. "Nonparametric Instrumental Variable Estimation of Binary Response Models with Continuous Endogenous Regressors," Econometrics and Statistics, Elsevier, vol. 17(C), pages 35-63.
    18. Hongjun Ha & Daniel Bauer, 2022. "A least-squares Monte Carlo approach to the estimation of enterprise risk," Finance and Stochastics, Springer, vol. 26(3), pages 417-459, July.
    19. Daniel Wilhelm, 2015. "Identification and estimation of nonparametric panel data regressions with measurement error," CeMMAP working papers CWP34/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    20. Irene Botosaru, 2017. "Identifying Distributions in a Panel Model with Heteroskedasticity: An Application to Earnings Volatility," Discussion Papers dp17-11, Department of Economics, Simon Fraser University.

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