Search Advanced

AIMS Mathematics

2022, Volume 7, Issue 8: 13704-13732. doi: 10.3934/math.2022755
Previous Article Next Article
Research article

Bootstrapping $ m $-generalized order statistics with variable rank

  • 1.
    Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
  • 2.
    Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt
  • 3.
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
  • Received: 21 January 2022 Revised: 11 April 2022 Accepted: 23 April 2022 Published: 23 May 2022

  • MSC : 62F20, 62F40, 62G30

  • In this paper, several bootstrap properties of $ m $-generalized order statistics ($ m $-GOSs) with variable rank (central and intermediate) are revealed. We study the inconsistency, weak consistency and strong consistency of bootstrapping central and intermediate $ m $-GOSs when the normalizing constants are assumed to be known or estimated from the re-sampled data using a proper re-sample size. Furthermore, sufficient conditions for the weak and strong consistencies of the bootstrapping distributions of central and intermediate $ m $-GOSs based on the normalizing constant estimators are given. Finally, a simulation study is conducted to determine the optimal bootstrap re-sample size corresponding to the best fitting of the bootstrapping distribution.

    Citation: H. M. Barakat, Magdy E. El-Adll, M. E. Sobh. Bootstrapping $ m $-generalized order statistics with variable rank[J]. AIMS Mathematics, 2022, 7(8): 13704-13732. doi: 10.3934/math.2022755

    Related Papers:

  • In this paper, several bootstrap properties of $ m $-generalized order statistics ($ m $-GOSs) with variable rank (central and intermediate) are revealed. We study the inconsistency, weak consistency and strong consistency of bootstrapping central and intermediate $ m $-GOSs when the normalizing constants are assumed to be known or estimated from the re-sampled data using a proper re-sample size. Furthermore, sufficient conditions for the weak and strong consistencies of the bootstrapping distributions of central and intermediate $ m $-GOSs based on the normalizing constant estimators are given. Finally, a simulation study is conducted to determine the optimal bootstrap re-sample size corresponding to the best fitting of the bootstrapping distribution.


    加载中


    [1] J. Adams, G. Atkinson, Development of seismic hazard maps for the proposed 2005 edition of the National Building Code of Canada, Can. J. Civil Eng., 30 (2003), 255–271. https://doi.org/10.1139/l02-070 doi: 10.1139/l02-070
    [2] K. B. Athreya, J. Fukuchi, Bootstrapping extremes of i.i.d random variables, In Proceedings of the Conference on Extreme Value Theory and Applications, NIST, Maryland, 7 (1994), 23–29.
    [3] K. B. Athreya, J. Fukuchi, Confidence interval for end point of a c.d.f, via bootstrap, J. Stat. Plan. Infer., 58 (1997), 299–320. https://doi.org/10.1016/S0378-3758(96)00087-0 doi: 10.1016/S0378-3758(96)00087-0
    [4] H. M. Barakat, Limit theory of generalized order statistics, J. Stat. Plan. Infer., 137 (2007), 1–11. https://doi.org/10.1016/j.jspi.2005.10.003 doi: 10.1016/j.jspi.2005.10.003
    [5] H. M. Barakat, M. A. El-Shandidy, Some limit theorems of intermediate term of a random number of independent random variables, Comment. Math. Univ. Ca., 31 (1990), 323–336. http://hdl.handle.net/10338.dmlcz/106862
    [6] H. M. Barakat, A. R. Omar, On limit distributions for intermediate order statistics under power normalization, Math. Methods Stat., 20 (2011), 365–377. https://doi.org/10.3103/S1066530711040053 doi: 10.3103/S1066530711040053
    [7] H. M. Barakat, A. R. Omar, On convergence of intermediate order statistics under power normalization, J. Stat. Appl. Pro., 4 (2015), 405–409. http://dx.doi.org/10.12785/jsap/040307 doi: 10.12785/jsap/040307
    [8] H. M. Barakat, M. E. El-Adll, A. E. Aly, Exact prediction intervals for future exponential lifetime based on random generalized order statistics, J. Comput. Math. Appl., 61 (2011), 1366–1378. https://doi.org/10.1016/j.camwa.2011.01.002 doi: 10.1016/j.camwa.2011.01.002
    [9] H. M. Barakat, E. M. Nigm, A. M. Alaswed, The Hill estimators under power normalization, Appl. Math. Model., 45 (2017), 813–822. https://doi.org/10.1016/j.apm.2017.01.028 doi: 10.1016/j.apm.2017.01.028
    [10] H. M. Barakat, E. M. Nigm, M. E. El-Adll, Bootstrap for extreme generalized order statistics, Arab. J. Sci. Eng., 36 (2011), 1083–1090. https://doi.org/10.1007/s13369-011-0105-1 doi: 10.1007/s13369-011-0105-1
    [11] H. M. Barakat, E. M. Nigm, M. H. Harpy, Limit theorems for univariate and bivariate order statistics with variable ranks, Statistics, 54 (2020), 737–755. https://doi.org/10.1080/02331888.2020.1772787 doi: 10.1080/02331888.2020.1772787
    [12] H. M. Barakat, E. M. Nigm, O. M. Khaled, F. A. Momenkhan, Bootstrap method for order statistics and modeling study of the air pollution, Commun. Stat.-Simul. C., 44 (2015), 1477–1491. https://doi.org/10.1080/03610918.2013.805051 doi: 10.1080/03610918.2013.805051
    [13] H. M. Barakat, E. M. Nigm, O. M. Khaled, Bootstrap method for central and intermediate order statistics under power normalization, Kybernetika, 51 (2015), 923–932. Available from: http://hdl.handle.net/10338.dmlcz/144817.
    [14] P. J. Bickel, D. A. Freedman, Some asymptotic theory for the bootstrap, Ann. Stat., 9 (1981), 1196–1217. https://doi.org/10.1214/aos/1176345637 doi: 10.1214/aos/1176345637
    [15] D. M. Chibisov, On limit distributions for order statistics, Theor. Probab. Appl., 9 (1964), 142–148. https://doi.org/10.1137/1109021 doi: 10.1137/1109021
    [16] E. W. Cope, G. Mignolia, G. Antonini, R. Ugoccioni, Challenges and pitfalls in measuring operational risk from loss data, J. Oper. Risk, 4 (2009), 3–27. https://doi.org/10.21314/JOP.2009.069 doi: 10.21314/JOP.2009.069
    [17] E. Cramer, Contributions to generalized order statistics, Habilitationsschrift, Reprint: University of Oldenburg, 2003.
    [18] L. de Haan, Fighting the arch-enemy with mathematics, Stat. Neerl., 44 (1990), 45–68. https://doi.org/10.1111/j.1467-9574.1990.tb01526.x doi: 10.1111/j.1467-9574.1990.tb01526.x
    [19] L. de Haan, A. Ferreira, Extreme value theory: An introduction, New York: Springer, 2006. https://doi.org/10.1007/0-387-34471-3
    [20] B. Efron, Bootstrap methods: Another look at the jackknife, Breakthroughs in statistics, New York: Springe, 1992,569–593. https://doi.org/10.1007/978-1-4612-4380-9_41
    [21] J. Fukuchi, Bootstrapping extremes of random variables, Doctoral dissertation, Ames, IA: Iowa State University, 1994. https://doi.org/10.31274/rtd-180813-10322
    [22] U. Kamps, A concept of generalized order statistics, J. Stat. Plan. Infer., 48 (1995), 1–23. https://doi.org/10.1016/0378-3758(94)00147-N doi: 10.1016/0378-3758(94)00147-N
    [23] M. R. Leadbetter, G. Lindgren, H. Rootzén, Extremes and related properties of random sequences and processes, New York: Springer, 1983. https://doi.org/10.1007/978-1-4612-5449-2
    [24] H. W. Lilliefors, On the Kolmogorov-Smirnov test for normality with mean and variance unknown, J. Am. Stat. Assoc., 62 (1967), 399–402. https://doi.org/10.1080/01621459.1967.10482916 doi: 10.1080/01621459.1967.10482916
    [25] D. M. Mason, Laws of large numbers for sums of extreme values, Ann. Probab., 10 (1982), 754–764. Available from: https://www.jstor.org/stable/2243383.
    [26] C. H. Nagaraja, H. N. Nagaraja, Distribution-free approximate methods for constructing confidence intervals for quantiles Int. Stat. Rev., 88 (2020), 75–100. https://doi.org/10.1111/insr.12338
    [27] D. Nasri-Roudsari, Extreme value theory of generalized order statistics, J. Stat. Plan. Inf., 55 (1996), 281–297. https://doi.org/10.1016/S0378-3758(95)00200-6 doi: 10.1016/S0378-3758(95)00200-6
    [28] D. Nasri-Roudsari, Limit distributions of generalized order statistics under power normalization, Commun. Stat.-Theor. M., 28 (1999), 1379–1389. https://doi.org/10.1080/03610929908832362 doi: 10.1080/03610929908832362
    [29] D. Nasri-Roudsari, E. Cramer, On the convergence rates of extreme generalized order statistics, Extremes, 2 (1999), 421–447. https://doi.org/10.1023/A:1009904316589 doi: 10.1023/A:1009904316589
    [30] J. Pickands Ⅲ, Statistical inference using extreme order statistics, Ann. Stat., 3 (1975), 119–131. https://www.jstor.org/stable/2958083
    [31] N. V. Smirnov, Limit distribution for terms of a variational series, Am. Math. Soc. Trans. Ser., 11 (1952), 82–143.
    [32] J. L. Teugels, Limit theorems on order statistics, Ann. Probab., 9 (1981), 868–880. Available from: https://www.jstor.org/stable/2243744.
    [33] C. Y. Wu, Types of limit distributions for some terms of variational series, Sci. Sin., 15(1966), 749–762.
  • Reader Comments
  • © 2022 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
2.2 3

Metrics

Article views(1120) PDF downloads(90) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(3)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog