Three-parameters Gumbel Distribution: Formulation and Parameter Estimation

Otieno Kevin Okumu; Omondi Joseph Ouno; Anthony Nyutu Karanjah; Samuel Nganga Muthiga

doi:doi:10.11648/j.ijsd.20241002.12

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Three-parameters Gumbel Distribution: Formulation and Parameter Estimation

Otieno Kevin Okumu^*, Omondi Joseph Ouno, Anthony Nyutu Karanjah, Samuel Nganga Muthiga

Published in International Journal of Statistical Distributions and Applications (Volume 10, Issue 2)

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

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[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.

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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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Author Information

Otieno Kevin Okumu

Department of Mathematics and Physical Sciences, School of Science, Maasai Mara University, Narok, Kenya

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Omondi Joseph Ouno

Department of Mathematics and Physical Sciences, School of Science, Maasai Mara University, Narok, Kenya

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Anthony Nyutu Karanjah

Department of Mathematics, Multimedia University, Nairobi, Kenya

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Samuel Nganga Muthiga

Department of Mathematics and Physical Sciences, School of Science, Maasai Mara University, Narok, Kenya

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Plain Text BibTeX RIS

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}
}

Copy | Download

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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