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Smooth conditional distribution estimators using Bernstein polynomials

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  • Belalia, Mohamed
  • Bouezmarni, Taoufik
  • Leblanc, Alexandre

Abstract

In a variety of statistical problems, estimation of the conditional distribution function remains a challenge. To this end, a two-stage Bernstein estimator for conditional distribution functions is introduced. The method consists in smoothing a first-stage Nadaraya–Watson or local linear estimator by constructing its Bernstein polynomial. Some asymptotic properties of the proposed estimator are derived, such as its asymptotic bias, variance and mean squared error. The asymptotic normality of the estimator is also established under appropriate conditions of regularity. Lastly, the performance of the proposed estimator is briefly studied through a few examples.

Suggested Citation

  • Belalia, Mohamed & Bouezmarni, Taoufik & Leblanc, Alexandre, 2017. "Smooth conditional distribution estimators using Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 111(C), pages 166-182.
    • Handle: RePEc:eee:csdana:v:111:y:2017:i:c:p:166-182
    DOI: 10.1016/j.csda.2017.02.005
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    References listed on IDEAS

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    1. Alexandre Leblanc, 2012. "On estimating distribution functions using Bernstein polynomials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 919-943, October.
    2. Alexandre Leblanc, 2010. "A bias-reduced approach to density estimation using Bernstein polynomials," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 459-475.
    3. Axel Tenbusch, 1994. "Two-dimensional Bernstein polynomial density estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 41(1), pages 233-253, December.
    4. Noël Veraverbeke & Irène Gijbels & Marek Omelka, 2014. "Preadjusted non-parametric estimation of a conditional distribution function," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(2), pages 399-438, March.
    5. Qi Li & Juan Lin & Jeffrey S. Racine, 2013. "Optimal Bandwidth Selection for Nonparametric Conditional Distribution and Quantile Functions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(1), pages 57-65, January.
    6. Hall, Peter & Wolff, Rodney C. L. & Yao, Qiwei, 1999. "Methods for estimating a conditional distribution function," LSE Research Online Documents on Economics 6631, London School of Economics and Political Science, LSE Library.
    7. Efromovich, Sam, 2010. "Dimension Reduction and Adaptation in Conditional Density Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 761-774.
    8. Babu, G. Jogesh & Chaubey, Yogendra P., 2006. "Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 959-969, May.
    9. Li, Qi & Racine, Jeffrey S, 2008. "Nonparametric Estimation of Conditional CDF and Quantile Functions With Mixed Categorical and Continuous Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 423-434.
    10. Belalia, Mohamed, 2016. "On the asymptotic properties of the Bernstein estimator of the multivariate distribution function," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 249-256.
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    Cited by:

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    3. Chen, Yaqing & Dawson, Matthew & Müller, Hans-Georg, 2020. "Rank dynamics for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 149(C).

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