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Degree selection methods for curve estimation via Bernstein polynomials

Author

Listed:
  • Juliana Freitas de Mello e Silva

    (Universidade Federal de Minas Gerais)

  • Sujit Kumar Ghosh

    (North Carolina State University)

  • Vinícius Diniz Mayrink

    (Universidade Federal de Minas Gerais)

Abstract

Bernstein Polynomial (BP) bases can uniformly approximate any continuous function based on observed noisy samples. However, a persistent challenge is the data-driven selection of a suitable degree for the BPs. In the absence of noise, asymptotic theory suggests that a larger degree leads to better approximation. However, in the presence of noise, which reduces bias, a larger degree also results in larger variances due to high-dimensional parameter estimation. Thus, a balance in the classic bias-variance trade-off is essential. The main objective of this work is to determine the minimum possible degree of the approximating BPs using probabilistic methods that are robust to various shapes of an unknown continuous function. Beyond offering theoretical guidance, the paper includes numerical illustrations to address the issue of determining a suitable degree for BPs in approximating arbitrary continuous functions.

Suggested Citation

  • Juliana Freitas de Mello e Silva & Sujit Kumar Ghosh & Vinícius Diniz Mayrink, 2025. "Degree selection methods for curve estimation via Bernstein polynomials," Computational Statistics, Springer, vol. 40(1), pages 1-26, January.
    • Handle: RePEc:spr:compst:v:40:y:2025:i:1:d:10.1007_s00180-024-01473-6
    DOI: 10.1007/s00180-024-01473-6
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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. Bruce M. Brown & Song Xi Chen, 1999. "Beta‐Bernstein Smoothing for Regression Curves with Compact Support," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(1), pages 47-59, March.
    3. Sonia Petrone, 1999. "Random Bernstein Polynomials," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 373-393, September.
    4. Qingning Zhou & Tao Hu & Jianguo Sun, 2017. "A Sieve Semiparametric Maximum Likelihood Approach for Regression Analysis of Bivariate Interval-Censored Failure Time Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 664-672, April.
    5. Yuhui Chen & Timothy Hanson & Jiajia Zhang, 2014. "Accelerated hazards model based on parametric families generalized with Bernstein polynomials," Biometrics, The International Biometric Society, vol. 70(1), pages 192-201, March.
    6. S. McKay Curtis & Sujit K. Ghosh, 2011. "A variable selection approach to monotonic regression with Bernstein polynomials," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(5), pages 961-976, February.
    7. I‐Shou Chang & Chao A. Hsiung & Yuh‐Jenn Wu & Che‐Chi Yang, 2005. "Bayesian Survival Analysis Using Bernstein Polynomials," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 447-466, September.
    8. 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.
    9. 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.
    10. Haiming Zhou & Timothy Hanson, 2018. "A Unified Framework for Fitting Bayesian Semiparametric Models to Arbitrarily Censored Survival Data, Including Spatially Referenced Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 571-581, April.
    11. Osman, Muhtarjan & Ghosh, Sujit K., 2012. "Nonparametric regression models for right-censored data using Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 559-573.
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