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Bernstein polynomial model for grouped continuous data

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  • Zhong Guan

Abstract

Grouped data are commonly encountered in applications. All data from a continuous population are grouped due to rounding of the individual observations. The Bernstein polynomial model is proposed as an approximate model in this paper for estimating a univariate density function based on grouped data. The coefficients of the Bernstein polynomial, as the mixture proportions of beta distributions, can be estimated using an EM algorithm. The optimal degree of the Bernstein polynomial can be determined using a change-point estimation method. The rate of convergence of the proposed density estimate to the true density is proved to be almost parametric by an acceptance–rejection argument used for generating random numbers. The proposed method is compared with some existing methods in a simulation study and is applied to the Chicken Embryo Data.

Suggested Citation

  • Zhong Guan, 2017. "Bernstein polynomial model for grouped continuous data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(4), pages 831-848, October.
    • Handle: RePEc:taf:gnstxx:v:29:y:2017:i:4:p:831-848
    DOI: 10.1080/10485252.2017.1374384
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    References listed on IDEAS

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    1. Wang, J. & Ghosh, S.K., 2012. "Shape restricted nonparametric regression with Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2729-2741.
    2. Nan Lin & Xuming He, 2006. "Robust and efficient estimation under data grouping," Biometrika, Biometrika Trust, vol. 93(1), pages 99-112, March.
    3. Zhong Guan, 2016. "Efficient and robust density estimation using Bernstein type polynomials," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 250-271, June.
    4. P. N. Jones & G. J. McLachlan, 1990. "Maximum Likelihood Estimation from Grouped and Truncated Data with Finite Normal Mixture Models," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 39(2), pages 273-282, June.
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    Cited by:

    1. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    2. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
    3. Wang, Tao & Guan, Zhong, 2023. "Choice of degree of Bernstein polynomial model," Statistics & Probability Letters, Elsevier, vol. 200(C).
    4. 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.
    5. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Papers 2011.00909, arXiv.org, revised Mar 2021.

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