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Distribution and density estimation based on variation-diminishing spline approximation

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  • Nezha Mohaoui

    (Faculty of Sciences Moulay Ismail University Zitoune)

  • Hamid Mraoui

    (Faculty of Sciences Mohammed First University Mohamed VI)

  • Abdelilah Monir

    (Faculty of Sciences Moulay Ismail University Zitoune)

Abstract

The estimation of distribution functions is a fundamental problem in the fields of statistics and machine learning. In this paper, based on Schoenberg’s variation-diminishing spline approximation, we propose an efficient nonparametric method for estimating distribution and density functions with bounded support. The preservation of the monotonicity of the distribution function and the positivity of the density function are guaranteed. Both methodology and asymptotic properties are established. We demonstrate theoretically and numerically that the smooth Schoenberg’s estimator can outperform the empirical cumulative distribution function. Several simulated examples and real data example are given to illustrate the efficiency and performance of our method. In the simulation study, the approach achieves very competitive performance with the kernel and the Bernstein polynomial estimators.

Suggested Citation

  • Nezha Mohaoui & Hamid Mraoui & Abdelilah Monir, 2025. "Distribution and density estimation based on variation-diminishing spline approximation," Statistical Papers, Springer, vol. 66(1), pages 1-31, January.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:1:d:10.1007_s00362-024-01619-0
    DOI: 10.1007/s00362-024-01619-0
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. A. Azzalini & A.W. Bowman, 1990. "A Look at Some Data on the Old Faithful Geyser," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 39(3), pages 357-365, November.
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