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Asymptotic Behavior of Multivariate Extremes Geometric Type Random Variables

Received: 14 September 2021     Accepted: 11 October 2021     Published: 11 November 2021
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Abstract

This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We succeeded in determining a subsequence which allows us to establish a convergence in law of the extremes of this type of random variable while passing by the determination of a speed of convergence. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type. These results will allow us to analyze, in a document, not only the convergence in moment of order of the other extremes of the random variables of geometric type but also the general asymptotic behavior of the extremes of a serie of random variables with integer value. This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We first made the case of the fact that the random variables of geometric type could be constructed from the random variables of exponential distribution and that they were not only integer variables but also that in general there were no sequences standards that allowed their extremes to converge. To do this, we first built a convergent ϕ(k) subsequence which we then used to define a geometric type Tϕ(k) subsequence of random variables. We have also proved the convergence in distribution of the extremes of the random variables Tϕ(k). We have also exhibited the resulting limit law. Finally, in this document, we have dealt with the multivariate case of random variables of geometric type. We considered two independent samples of random variables of geometric types. Using a copula of extreme values, in particular the logistic copula, we proposed a joint limit distribution of two independent samples of subsequences of geometric type random variables. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type.

Published in Applied and Computational Mathematics (Volume 10, Issue 6)
DOI 10.11648/j.acm.20211006.12
Page(s) 138-145
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Asymptotic Convergence, Generalized Extreme Value Distribution, Exponantial and Geometric Distribution, Extreme Values Copulas

References
[1] A. V. Skorokhod Limit Theorems for Stochastic Processes Theory Probab. Appl., 1956, Vol. 1, No. 3: pp. 261-290, Society for Industrial and Applied Mathematics
[2] A. V. Stepanov Limits theorem for weak records, 1990.
[3] C. Genest, J. Segers, Rank-based inference for bivariate extreme-value copulas , Ann. Stat. 37 (2009), No 5B, p. 2990-3022.
[4] C. W. Anderson Extreme value theory for a class of discrete distributions with applications to some stochastique processe, J.Appl. Pron. No 7, pp 99-113, 1970.
[5] F. Thomas Bruss, Rudolf Grübel, On the multiplicity of the maximum in a discrete random sample, The annals of Applied Probability, vol.13, No 4, pp 1252-1263, 2003.
[6] J. Galambos, The asymptotic theory of extreme order statistics, juin 1978, pp 85.
[7] J. Pickands, Moment convergence of sample extremes, Ann. Math. Statist. 39 (1968), 881-889.
[8] K. Ghoudi, A. Khoudraji, L.-P. Rivest, Propriétés statistiques des copules de valeurs extrˆ emes bidimensionnelles, Can. J. Stat. 26 (1998), No 1, p. 187-197.
[9] R. A. Fisher et L. H. C. Tippet, Limiting forms of frequency distribution of the largest or smallest member of a sample, Mathematical Proceedings of the Cambridge Philosophical Society, volume 24, numéro 2, Acril 1928, pp. 180-190.
[10] S. I. Resnick, Extreme values, Regular variation and Point processes, Springer-Verlag, New york, 1987, pp. 443-45.
[11] T. Bastogne, R.Keinj and P.Vallois Multinomial model- based formulations of tcp and ntcp for radiotherapy treatment planning. Journal of Theoretical Biology, 279: 55-62, 2011.
[12] W. J. Hall, Jon A. Wellner Estimation of Mean Residual Life, Statistical Modeling for Biological Systems pp 169- 189, jul 2017.
[13] Z. Peng and S. Nadarajah Convergence rate for the moments of extremes, Bull. Korean Math. Soc. 49(2012) No. 3, pp. 495-510.
[14] Tiago de Olivera, Extremal distributions Rev. Fac. Ciencas Univ. Lisboa, A8, 299-310, 1958.
[15] Sibuya, M. Bivariate extreme statistics Ann. Inst. Statist. Math., 11, 195-210, 1960.
[16] Geoffroy, J. Contribution B la thtorie des valeurs extremes Publ. Inst. Statist. Univ. Paris, 7, 37-185, 1958.
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    Frédéric Béré, Kpèbbèwèrè Cédric Somé, Remi Guillaume Bagré, Pierre Clovis Nitiéma. (2021). Asymptotic Behavior of Multivariate Extremes Geometric Type Random Variables. Applied and Computational Mathematics, 10(6), 138-145. https://doi.org/10.11648/j.acm.20211006.12

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    ACS Style

    Frédéric Béré; Kpèbbèwèrè Cédric Somé; Remi Guillaume Bagré; Pierre Clovis Nitiéma. Asymptotic Behavior of Multivariate Extremes Geometric Type Random Variables. Appl. Comput. Math. 2021, 10(6), 138-145. doi: 10.11648/j.acm.20211006.12

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    AMA Style

    Frédéric Béré, Kpèbbèwèrè Cédric Somé, Remi Guillaume Bagré, Pierre Clovis Nitiéma. Asymptotic Behavior of Multivariate Extremes Geometric Type Random Variables. Appl Comput Math. 2021;10(6):138-145. doi: 10.11648/j.acm.20211006.12

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  • @article{10.11648/j.acm.20211006.12,
      author = {Frédéric Béré and Kpèbbèwèrè Cédric Somé and Remi Guillaume Bagré and Pierre Clovis Nitiéma},
      title = {Asymptotic Behavior of Multivariate Extremes Geometric Type Random Variables},
      journal = {Applied and Computational Mathematics},
      volume = {10},
      number = {6},
      pages = {138-145},
      doi = {10.11648/j.acm.20211006.12},
      url = {https://doi.org/10.11648/j.acm.20211006.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211006.12},
      abstract = {This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We succeeded in determining a subsequence which allows us to establish a convergence in law of the extremes of this type of random variable while passing by the determination of a speed of convergence. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type. These results will allow us to analyze, in a document, not only the convergence in moment of order of the other extremes of the random variables of geometric type but also the general asymptotic behavior of the extremes of a serie of random variables with integer value. This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We first made the case of the fact that the random variables of geometric type could be constructed from the random variables of exponential distribution and that they were not only integer variables but also that in general there were no sequences standards that allowed their extremes to converge. To do this, we first built a convergent ϕ(k) subsequence which we then used to define a geometric type Tϕ(k) subsequence of random variables. We have also proved the convergence in distribution of the extremes of the random variables Tϕ(k). We have also exhibited the resulting limit law. Finally, in this document, we have dealt with the multivariate case of random variables of geometric type. We considered two independent samples of random variables of geometric types. Using a copula of extreme values, in particular the logistic copula, we proposed a joint limit distribution of two independent samples of subsequences of geometric type random variables. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type.},
     year = {2021}
    }
    

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    AU  - Frédéric Béré
    AU  - Kpèbbèwèrè Cédric Somé
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    JO  - Applied and Computational Mathematics
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    UR  - https://doi.org/10.11648/j.acm.20211006.12
    AB  - This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We succeeded in determining a subsequence which allows us to establish a convergence in law of the extremes of this type of random variable while passing by the determination of a speed of convergence. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type. These results will allow us to analyze, in a document, not only the convergence in moment of order of the other extremes of the random variables of geometric type but also the general asymptotic behavior of the extremes of a serie of random variables with integer value. This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We first made the case of the fact that the random variables of geometric type could be constructed from the random variables of exponential distribution and that they were not only integer variables but also that in general there were no sequences standards that allowed their extremes to converge. To do this, we first built a convergent ϕ(k) subsequence which we then used to define a geometric type Tϕ(k) subsequence of random variables. We have also proved the convergence in distribution of the extremes of the random variables Tϕ(k). We have also exhibited the resulting limit law. Finally, in this document, we have dealt with the multivariate case of random variables of geometric type. We considered two independent samples of random variables of geometric types. Using a copula of extreme values, in particular the logistic copula, we proposed a joint limit distribution of two independent samples of subsequences of geometric type random variables. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type.
    VL  - 10
    IS  - 6
    ER  - 

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Author Information
  • Department of Mathematics, Institute of Sciences, Ouagadougou, Burkina Faso

  • Department of Mathematics, Virtual University of Burkina Faso, Ouagadougou, Burkina Faso

  • Department of Mathematics, Norbert ZONGO University, Koudougou, Burkina Faso

  • Department of Mathematics of Decision, Thomas SANKARA University, Ouagadougou, Burkina Faso

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