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Efficient and robust density estimation using Bernstein type polynomials

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  • Zhong Guan

Abstract

A method of parameterising and smoothing the unknown underlying distributions using Bernstein type polynomials with positive coefficients is proposed, verified and investigated. Any distribution with bounded and smooth enough density can be approximated by the proposed model which turns out to be a mixture of the beta distributions, beta , , for some optimal degree m . A simple change-point estimating method for choosing the optimal degree m of the approximate model is presented. The proposed method gives a maximum likelihood density estimate which is consistent in distance at a nearly parametric rate under some conditions. Simulation study shows that one can benefit from both the smoothness and the efficiency by using the proposed method which can also be used to estimate some population parameters such as the mean. The proposed methods are applied to three data sets of different types.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:gnstxx:v:28:y:2016:i:2:p:250-271
    DOI: 10.1080/10485252.2016.1163349
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    Cited by:

    1. Zhong Guan, 2017. "Bernstein polynomial model for grouped continuous data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(4), pages 831-848, October.
    2. Aurélie Bertrand & Ingrid Van Keilegom & Catherine Legrand, 2019. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," Biometrics, The International Biometric Society, vol. 75(1), pages 297-307, March.
    3. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    4. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
    5. Wang, Tao & Guan, Zhong, 2023. "Choice of degree of Bernstein polynomial model," Statistics & Probability Letters, Elsevier, vol. 200(C).
    6. Cheng, Xueqin & Li, Yanpeng, 2022. "An improved Hoeffding’s inequality for sum of independent random variables," Statistics & Probability Letters, Elsevier, vol. 183(C).
    7. Bertrand, Aurelie & Van Keilegom, Ingrid & Legrand, Catherine, 2017. "Flexible parametric approach to classical measurement error variance estimation without auxiliary data," LIDAM Discussion Papers ISBA 2017025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. 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.
    9. Dietmar Pfeifer & Olena Ragulina, 2020. "Adaptive Bernstein Copulas and Risk Management," Papers 2011.00909, arXiv.org, revised Mar 2021.
    10. Dongliang Wang & Xueya Cai, 2021. "Smooth ROC curve estimation via Bernstein polynomials," PLOS ONE, Public Library of Science, vol. 16(5), pages 1-12, May.

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