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Application of Bernstein Polynomials on Estimating a Distribution and Density Function in a Triangular Array

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  • Lina Wang

    (Dalian University of Technology)

  • Dawei Lu

    (Dalian University of Technology)

Abstract

In this paper, we study some asymptotic properties for the Bernstein estimators of the limit distribution function and the limit density function under a triangular sample. Specifically, we obtain the uniform strong consistency, mean squared error (MSE) and mean integrated squared error (MISE) for the resulting estimators. In addition, we give the optimal choice of the bandwidth parameter m in terms of the sample size n, for both the MSE and MISE. Numerical simulations are presented to show that the Bernstein estimators outperform Gaussian kernel estimators in terms of MISE under a triangular sample.

Suggested Citation

  • Lina Wang & Dawei Lu, 2023. "Application of Bernstein Polynomials on Estimating a Distribution and Density Function in a Triangular Array," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-14, June.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:2:d:10.1007_s11009-023-10032-3
    DOI: 10.1007/s11009-023-10032-3
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    References listed on IDEAS

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    1. Dawei Lu & Lina Wang, 2021. "On the Rates of Asymptotic Normality for Bernstein Polynomial Estimators in a Triangular Array," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1519-1536, December.
    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. Gaku Igarashi & Yoshihide Kakizawa, 2014. "On improving convergence rate of Bernstein polynomial density estimator," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(1), pages 61-84, March.
    4. Dawei Lu & Lina Wang & Jingcai Yang, 2022. "The stochastic convergence of Bernstein polynomial estimators in a triangular array," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 34(4), pages 987-1014, October.
    5. 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.
    6. 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.
    7. 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.
    8. Galen R. Shorack, 1979. "The weighted empirical process of row independent random variables with arbitrary distribution functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 33(4), pages 169-189, December.
    9. 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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