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Consistency and asymptotic normality for a nonparametric prediction under measurement errors

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  • Mynbaev, Kairat
  • Martins-Filho, Carlos

Abstract

Nonparametric prediction of a random variable Y conditional on the value of an explanatory variable X is a classical and important problem in Statistics. The problem is significantly complicated if there are heterogeneously distributed measurement errors on the observed values of X used in estimation and prediction. Carroll et al. (2009) have recently proposed a kernel deconvolution estimator and obtained its consistency. In this paper we use the kernels proposed in Mynbaev and Martins-Filho (2010) to define a class of deconvolution estimators for prediction that contains their estimator as one of its elements. First, we obtain consistency of the estimators under much less restrictive conditions. Specifically, contrary to what is routinely assumed in the extant literature, the Fourier transform of the underlying kernels is not required to have compact support, higher-order restrictions on the kernel can be avoided and fractional smoothness of the involved densities is allowed. Second, we obtain asymptotic normality of the estimators under the assumption that there are two types of measurement errors on the observed values of X. It is apparent from our study that even in this simplified setting there are multiple cases exhibiting different asymptotic behavior. Our proof focuses on the case where measurement errors are super-smooth and we use it to discuss other possibilities. The results of a Monte Carlo simulation are provided to compare the performance of the estimator using traditional kernels and those proposed in Mynbaev and Martins-Filho (2010).

Suggested Citation

  • Mynbaev, Kairat & Martins-Filho, Carlos, 2015. "Consistency and asymptotic normality for a nonparametric prediction under measurement errors," MPRA Paper 75845, University Library of Munich, Germany, revised 2014.
  • Handle: RePEc:pra:mprapa:75845
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    References listed on IDEAS

    as
    1. Kairat Mynbaev & Carlos Martins-Filho, 2010. "Bias reduction in kernel density estimation via Lipschitz condition," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(2), pages 219-235.
    2. Masry, E., 1993. "Asymptotic Normality for Deconvolution Estimators of Multivariate Densities of Stationary Processes," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 47-68, January.
    3. Toshio Honda, 2009. "Nonparametric regression for dependent data in the errors-in-variables problem," Global COE Hi-Stat Discussion Paper Series gd09-092, Institute of Economic Research, Hitotsubashi University.
    4. Mynbaev, Kairat, 2011. "Distributions escaping to infinity and the limiting power of the Cliff-Ord test for autocorrelation," MPRA Paper 44402, University Library of Munich, Germany, revised 18 Sep 2012.
    5. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 471-480, September.
    6. Delaigle, Aurore & Meister, Alexander, 2007. "Nonparametric Regression Estimation in the Heteroscedastic Errors-in-Variables Problem," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1416-1426, December.
    7. Fan, Jianqing & Masry, Elias, 1992. "Multivariate regression estimation with errors-in-variables: Asymptotic normality for mixing processes," Journal of Multivariate Analysis, Elsevier, vol. 43(2), pages 237-271, November.
    8. Bert Van Es & Hae‐Won Uh, 2005. "Asymptotic Normality of Kernel‐Type Deconvolution Estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 467-483, September.
    9. Carroll, Raymond J. & Delaigle, Aurore & Hall, Peter, 2009. "Nonparametric Prediction in Measurement Error Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 993-1003.
    10. Staudenmayer, John & Ruppert, David & Buonaccorsi, John P., 2008. "Density Estimation in the Presence of Heteroscedastic Measurement Error," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 726-736, June.
    11. Cator, Eric A., 2001. "Deconvolution with arbitrarily smooth kernels," Statistics & Probability Letters, Elsevier, vol. 54(2), pages 205-214, September.
    12. Delaigle, Aurore & Fan, Jianqing & Carroll, Raymond J., 2009. "A Design-Adaptive Local Polynomial Estimator for the Errors-in-Variables Problem," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 348-359.
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    More about this item

    Keywords

    Measurement errors; nonparametric prediction; asymptotic normality; Lipschitz conditions;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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