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Three-parameters Gumbel Distribution: Formulation and Parameter Estimation

Received: 6 March 2024     Accepted: 3 April 2024     Published: 20 June 2024
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Abstract

Modeling extreme value theories is really gaining interest in the world with scientist working to improve the flexibility of the distributions by adding parameter(s). Extreme value distributions are always described to include families of Gumbel, Weibull and Frechet distributions. Of the three distributions, Gumbel distribution is the most commonly used in the extreme value theory analysis. Existing literature has shown that the addition of parameter to a distribution makes it robust and/or more flexible hence the study intends to improve the existing two parameters Gumbel distribution using the Marshall and Olkin proposed method for introducing a new estimator/parameter to an existing distribution. The developed distribution will be important to the applications in some life time studies like high temperature, earthquakes, network designs, horse racing, queues in supermarket, insurance, winds, risk management, ozone concentration, flood, engineering and financial concepts. The parameters for the introduced distribution was estimated using Maximum Likelihood Estimation method. The introduced three parameters Gumbel distribution is a probability distribution function which can be used in modelling statistical data. The maximum likelihood estimates for the three parameters namely shape, location and dispersion are efficient, sufficient and consistent and this makes the function more flexible and better for application. The three parameters Gumbel distribution can be used in modeling and analysis of normal data, skewed data and extreme data since it will provide efficient, sufficient and consistent estimates.

Published in International Journal of Statistical Distributions and Applications (Volume 10, Issue 2)
DOI 10.11648/j.ijsd.20241002.12
Page(s) 25-37
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), 2024. Published by Science Publishing Group

Keywords

Gumbel Distribution, Maximum Likelihood Estimation, Marshall Olkins Method, Parameters, Three Parameters Gumbel Distribution

References
[1] Abdelaziz, Q. and Zoglat, A. (2011). Discriminating between normal and Gumbel distributions.
[2] Abouelmagd, T. H. M., and Hamed, M. S. (2018). The Burr X Frechet distribution with its properties and applications. Journal of Applied Probability and Statistics, 13(1): 23-51.
[3] Afify, A. Z., and Zayed, M. (2018). The extended exponential distribution and its application. Journal of Statistical Theory and Applications, 17(2): 213-229.
[4] Anwar, H., Dar, I. H., and Lone, M. A. (2022). A novel family of generating distributions based on Marshall Olkin transformation with an application to exponential distribution. Pak. J. Statist, 38(1): 113-128.
[5] Coles, S. (2001).An introduction to statistical modeling of extreme values. Springer-Verlag.
[6] Ekramy, A. H., Aljohani, H. M., and Ahmed, Z. A. (2021). The extended Weibull-Frechet distribution: properties, inference, and application in medicine and engineering. AIMS Mathematics, 7(1): 225-246.
[7] Garg, M., Gupta, J., and Gupta, M., (2016). The Lomax Gumbel. Palestine journal of Mathematics, 5(1): 35-42.
[8] Gumbel, E. J. (1958). Statistics of extreme, Colombia University Press.
[9] Gupta, R., and Biswas, A. (2010). Wind data analysis of Silchar (Assam India) by Rayleigh and Weibull methods: Journal of mechanical engineering research, 2: 10-24.
[10] Hurlin, C., (2013). Maximum likelihood estimation. Advanced econometrics, University of Orleans.
[11] Khalil, G. M., and Rezk, H. (2019). Extended poison Frechet distribution and its application, 15(4): 905-919.
[12] Klara, P., and Jesper, R. (2010). Exponentiated Gumbel distribution for estimation of return levels of significant wave heights. Journal of environmental statistics, 1(3).
[13] Lawan, S. M., Abidin, W. A. W. Z., Chai, W. Y., Baharum, A., and Masri, T., (2015). Statistical modelling of long-term wind speed data : American journal of computer science and information technology, 13: 79- 121.
[14] Neamah, M. W., and Qasim, B. A., (2021). A new truncated Gumbel distribution: Properties and Estimation. Journal of Physics: Conference series
[15] Pedro, L. R., Nascimento, D., and Louzada, F. (2016). The Long Term Frechet distribution: Estimation, Properties and its Applications. Institute of Mathematical Science and Computing, University of Sao Paulo.
[16] Pericchi, L., and Coles, S. (2003). Anticipating catastrophes through extreme value modeling. Appl. Statist. 52: 405-416.
[17] Phien, N. H., and Fang, E. T. (1989). Parameter estimation for the general extreme value distribution, 15(2).
[18] Subh, S. A., and Alodat, M. T. (2017). Discrimination between Logistic and Gumbel distribution. International Journal of Sciences: Basic and Applied Research, 36(3): 244-255.
[19] Teamah, A. M. A., Elbanna, A. A., and Gemeay, A. M. (2020). Frechet-Weibull distribution with applications to earthquakes data sets. Pak. J. Statist, 36(2): 135-147.
[20] Zheng, S., (2018). Maximum Likelihood Estimation. Statistical theory II.
Cite This Article
  • APA Style

    Okumu, O. K., Ouno, O. J., Karanjah, A. N., Muthiga, S. N. (2024). Three-parameters Gumbel Distribution: Formulation and Parameter Estimation. International Journal of Statistical Distributions and Applications, 10(2), 25-37. https://doi.org/10.11648/j.ijsd.20241002.12

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    ACS Style

    Okumu, O. K.; Ouno, O. J.; Karanjah, A. N.; Muthiga, S. N. Three-parameters Gumbel Distribution: Formulation and Parameter Estimation. Int. J. Stat. Distrib. Appl. 2024, 10(2), 25-37. doi: 10.11648/j.ijsd.20241002.12

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    AMA Style

    Okumu OK, Ouno OJ, Karanjah AN, Muthiga SN. Three-parameters Gumbel Distribution: Formulation and Parameter Estimation. Int J Stat Distrib Appl. 2024;10(2):25-37. doi: 10.11648/j.ijsd.20241002.12

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  • @article{10.11648/j.ijsd.20241002.12,
      author = {Otieno Kevin Okumu and Omondi Joseph Ouno and Anthony Nyutu Karanjah and Samuel Nganga Muthiga},
      title = {Three-parameters Gumbel Distribution: Formulation and Parameter Estimation},
      journal = {International Journal of Statistical Distributions and Applications},
      volume = {10},
      number = {2},
      pages = {25-37},
      doi = {10.11648/j.ijsd.20241002.12},
      url = {https://doi.org/10.11648/j.ijsd.20241002.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20241002.12},
      abstract = {Modeling extreme value theories is really gaining interest in the world with scientist working to improve the flexibility of the distributions by adding parameter(s). Extreme value distributions are always described to include families of Gumbel, Weibull and Frechet distributions. Of the three distributions, Gumbel distribution is the most commonly used in the extreme value theory analysis. Existing literature has shown that the addition of parameter to a distribution makes it robust and/or more flexible hence the study intends to improve the existing two parameters Gumbel distribution using the Marshall and Olkin proposed method for introducing a new estimator/parameter to an existing distribution. The developed distribution will be important to the applications in some life time studies like high temperature, earthquakes, network designs, horse racing, queues in supermarket, insurance, winds, risk management, ozone concentration, flood, engineering and financial concepts. The parameters for the introduced distribution was estimated using Maximum Likelihood Estimation method. The introduced three parameters Gumbel distribution is a probability distribution function which can be used in modelling statistical data. The maximum likelihood estimates for the three parameters namely shape, location and dispersion are efficient, sufficient and consistent and this makes the function more flexible and better for application. The three parameters Gumbel distribution can be used in modeling and analysis of normal data, skewed data and extreme data since it will provide efficient, sufficient and consistent estimates.},
     year = {2024}
    }
    

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  • TY  - JOUR
    T1  - Three-parameters Gumbel Distribution: Formulation and Parameter Estimation
    AU  - Otieno Kevin Okumu
    AU  - Omondi Joseph Ouno
    AU  - Anthony Nyutu Karanjah
    AU  - Samuel Nganga Muthiga
    Y1  - 2024/06/20
    PY  - 2024
    N1  - https://doi.org/10.11648/j.ijsd.20241002.12
    DO  - 10.11648/j.ijsd.20241002.12
    T2  - International Journal of Statistical Distributions and Applications
    JF  - International Journal of Statistical Distributions and Applications
    JO  - International Journal of Statistical Distributions and Applications
    SP  - 25
    EP  - 37
    PB  - Science Publishing Group
    SN  - 2472-3509
    UR  - https://doi.org/10.11648/j.ijsd.20241002.12
    AB  - Modeling extreme value theories is really gaining interest in the world with scientist working to improve the flexibility of the distributions by adding parameter(s). Extreme value distributions are always described to include families of Gumbel, Weibull and Frechet distributions. Of the three distributions, Gumbel distribution is the most commonly used in the extreme value theory analysis. Existing literature has shown that the addition of parameter to a distribution makes it robust and/or more flexible hence the study intends to improve the existing two parameters Gumbel distribution using the Marshall and Olkin proposed method for introducing a new estimator/parameter to an existing distribution. The developed distribution will be important to the applications in some life time studies like high temperature, earthquakes, network designs, horse racing, queues in supermarket, insurance, winds, risk management, ozone concentration, flood, engineering and financial concepts. The parameters for the introduced distribution was estimated using Maximum Likelihood Estimation method. The introduced three parameters Gumbel distribution is a probability distribution function which can be used in modelling statistical data. The maximum likelihood estimates for the three parameters namely shape, location and dispersion are efficient, sufficient and consistent and this makes the function more flexible and better for application. The three parameters Gumbel distribution can be used in modeling and analysis of normal data, skewed data and extreme data since it will provide efficient, sufficient and consistent estimates.
    VL  - 10
    IS  - 2
    ER  - 

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