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The empirical Bernstein process with application to uniformity testing

Author

Listed:
  • Ran Sun

    (University of Windsor)

  • Mohamed Belalia

    (University of Windsor)

  • Sévérien Nkurunziza

    (University of Windsor)

Abstract

In this study, we introduce empirical Bernstein process and establish its weak convergence. We also present a novel testing procedure for assessing uniformity, which utilizes the Cramér–Von Mises and Kolmogorov–Smirnov functionals of the empirical Bernstein process. Additionally, we derive the asymptotic properties of the proposed tests’ statistics under the null hypothesis and under a sequence of local alternative hypotheses. Comprehensive simulation studies demonstrate that tests outperform those based solely on the empirical distribution.

Suggested Citation

  • Ran Sun & Mohamed Belalia & Sévérien Nkurunziza, 2025. "The empirical Bernstein process with application to uniformity testing," Statistical Papers, Springer, vol. 66(2), pages 1-25, February.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:2:d:10.1007_s00362-025-01668-z
    DOI: 10.1007/s00362-025-01668-z
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    References listed on IDEAS

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    1. Alexandre Leblanc, 2009. "Chung–Smirnov property for Bernstein estimators of distribution functions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 133-142.
    2. Babu, G. Jogesh & Chaubey, Yogendra P., 2006. "Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 959-969, May.
    3. 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.
    4. 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.
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