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AIMS Mathematics

2026, Volume 11, Issue 8: 26308-26336. doi: 10.3934/math.20261056
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Research article

Asymptotic behavior of bootstrapped extreme $ m $-generalized order statistics with application to left endpoint estimation

  • 1.
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
  • 2.
    Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
  • 3.
    Department of Mathematics, Faculty of Science, Capital University (Formerly Helwan University), Cairo, Egypt
  • 4.
    Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
  • Published: 24 August 2026

  • MSC : Primary 62G32; Secondary 60F05, 62G09, 62G20, 62G30

  • This paper develops an asymptotic theory for the bootstrapped distributions of normalized upper and lower extreme $ m $-generalized order statistics ($ m $-GOSes). Sufficient conditions are established for their weak convergence when the linear normalizing constants are unknown and must be estimated. The proposed asymptotic framework is developed under suitable consistency conditions for the estimators of the normalizing constants together with the asymptotic requirement that the bootstrap sample size satisfies $ M = o(n). $ As an application, asymptotic bootstrap confidence intervals for the left endpoint of distributions in the Weibull domain of attraction are constructed using a pivotal quantity based on lower extreme $ m $-GOSes. The proposed procedure is applicable to both original and sequential order statistics models. A Monte Carlo simulation study is conducted to evaluate the performance of constructing bootstrap confidence intervals for the left endpoints of both exponential and gamma distributions. The results reveal that the bootstrap method provides robust inference whenever the theoretical assumptions are met, which supports the use of bootstrap procedures for reliable estimation of the lower endpoint in finite samples.

    Citation: M. E. Sobh, H. M. Barakat, Magdy E. El-Adll, Amany E. Aly, Asamh Saleh M. Al Luhayb. Asymptotic behavior of bootstrapped extreme $ m $-generalized order statistics with application to left endpoint estimation[J]. AIMS Mathematics, 2026, 11(8): 26308-26336. doi: 10.3934/math.20261056

    Related Papers:

  • This paper develops an asymptotic theory for the bootstrapped distributions of normalized upper and lower extreme $ m $-generalized order statistics ($ m $-GOSes). Sufficient conditions are established for their weak convergence when the linear normalizing constants are unknown and must be estimated. The proposed asymptotic framework is developed under suitable consistency conditions for the estimators of the normalizing constants together with the asymptotic requirement that the bootstrap sample size satisfies $ M = o(n). $ As an application, asymptotic bootstrap confidence intervals for the left endpoint of distributions in the Weibull domain of attraction are constructed using a pivotal quantity based on lower extreme $ m $-GOSes. The proposed procedure is applicable to both original and sequential order statistics models. A Monte Carlo simulation study is conducted to evaluate the performance of constructing bootstrap confidence intervals for the left endpoints of both exponential and gamma distributions. The results reveal that the bootstrap method provides robust inference whenever the theoretical assumptions are met, which supports the use of bootstrap procedures for reliable estimation of the lower endpoint in finite samples.



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  • © 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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