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On improving convergence rate of Bernstein polynomial density estimator

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  • Gaku Igarashi
  • Yoshihide Kakizawa

Abstract

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), 'A Bernstein Polynomial Approach to Density Function Estimation', in Statistical Inference and Related Topics , ed. M.L. Puri, 2, New York: Academic Press, pp. 87-99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), 'On Improving Convergence Rates for Nonnegative Kernel Density Estimators', The Annals of Statistics , 8, 1160-1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), 'A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials', Journal of Nonparametric Statistics , 22, 459-475] are shown to be O ( n -super- - 8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O ( n -super- - 8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:gnstxx:v:26:y:2014:i:1:p:61-84
    DOI: 10.1080/10485252.2013.827195
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    References listed on IDEAS

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    1. 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.
    2. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    3. Hirukawa, Masayuki, 2010. "Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 473-495, February.
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    Cited by:

    1. Lu, Lu, 2015. "On the uniform consistency of the Bernstein density estimator," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 52-61.
    2. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    3. Igarashi, Gaku & Kakizawa, Yoshihide, 2014. "Re-formulation of inverse Gaussian, reciprocal inverse Gaussian, and Birnbaum–Saunders kernel estimators," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 235-246.
    4. Ouimet, Frédéric & Tolosana-Delgado, Raimon, 2022. "Asymptotic properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    5. 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.
    6. Frédéric Ouimet, 2021. "General Formulas for the Central and Non-Central Moments of the Multinomial Distribution," Stats, MDPI, vol. 4(1), pages 1-10, January.

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