Inferences on the Weibull Exponentiated Exponential Distribution and Applications

Umar Usman; Suleiman Shamsuddeen; Bello Magaji Arkilla; Yakubu Aliyu

doi:doi:10.11648/j.ijsd.20200601.12

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Inferences on the Weibull Exponentiated Exponential Distribution and Applications

Umar Usman, Suleiman Shamsuddeen, Bello Magaji Arkilla, Yakubu Aliyu

Published in International Journal of Statistical Distributions and Applications (Volume 6, Issue 1)

Received: 6 November 2019 Accepted: 20 December 2019 Published: 15 July 2020

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Abstract

In this article, an alternative method of defining the probability density function of GeneralizedWeibull-exponential distributions is proposed. Based on the method, the distribution can also be calledWeibull exponentiated exponential distribution. This distribution includes the exponential, Weibull and exponentiated exponential distributions as special cases. Comprehensive mathematical treatment of the distribution is provided. The quantile function, mode, characteristic function, moment generating function among other mathematical properties of the distribution were derived. The parameters of the distribution were estimated by applying the Maximum Likelihood Procedure.The elements of the Fisher Information Matrix is also provided. Finally, a data set is fitted to the model and its sub-models. It is observed that the new distribution is more flexible and can be used quiet effectively in analysing real life data in place of exponential, Weibull and exponentiated exponential distributions.

Published in	International Journal of Statistical Distributions and Applications (Volume 6, Issue 1)
DOI	10.11648/j.ijsd.20200601.12
Page(s)	10-22
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), 2020. Published by Science Publishing Group

Keywords

T-X Family, Exponentiated Exponential Distribution, Order Statistics, Shannon Entropy and Likelihood Ratio Test

References

[1]	M. A. E. M. A. Elgarhy, Contributions to a Family of Generated Distributions. PhD thesis, Institute of Statistical Studies and Research, Cairo University, 2017.
[2]	N. Eugene, C. Lee, and F. Famoye, “Beta-normal distribution and its applications,” Communications in Statistics-Theory and methods, vol. 31, no. 4, pp. 497–512, 2002.
[3]	M. M. Mansour, M. S. Hamed, and S. M. Mohamed, “A new kumaraswamy transmuted modified weibull distribution with application,” J. Stat. Adv. Theory Applic, vol. 13, pp. 101–133, 2015.
[4]	W. Barreto-Souza, A. H. Santos, and G. M. Cordeiro, “The beta generalized exponential distribution,” Journal of Statistical Computation and Simulation, vol. 80, no. 2, pp. 159–172, 2010.
[5]	H. G. Dikko, Y. Aliyu, and S. Alfa, “The beta-burr type v distribution: its properties and application to real life data,” International Journal of Advanced Statistics and Probability, 5 (2), vol. 83, 2017.
[6]	A. Usman, A. Ishaq, M. Tasi´u, and Y. Aliyu, “Weibullburr type x distribution: Its properties and application,” Nigerian Journal of Scientific Research, vol. 16, no. 1, pp. 150–157, 2017.
[7]	A. Yakubu and S. I. Doguwa, “On the properties of the weibull-burr iii distribution and its application to uncensored and censored survival data,” CBN Journal of Applied Statistics, vol. 8, no. 2, pp. 91–116, 2017.
[8]	H. M. Salem and M. Selim, “The generalized weibullexponential distribution: properties and applications,” International Journal of Statistics and Applications, vol. 4, no. 2, pp. 102–112, 2014.
[9]	A. Alzaghal, F. Famoye, and C. Lee, “Exponentiated tx family of distributions with some applications,” International Journal of Statistics and Probability, vol. 2, no. 3, p. 31, 2013.
[10]	A. Alzaatreh, C. Lee, and F. Famoye, “A new method for generating families of continuous distributions,” Metron, vol. 71, no. 1, pp. 63–79, 2013.
[11]	R. D. Gupta and D. Kundu, “Theory & methods: Generalized exponential distributions,” Australian & New Zealand Journal of Statistics, vol. 41, no. 2, pp. 173–188, 1999.
[12]	R. D. Gupta and D. Kundu, “Generalized exponential distribution: An alternative to gamma and weibull distributions,” Biom. J, vol. 43, no. 1, pp. 117–130, 2001a.
[13]	D. Kundu and R. Gupta, “Generalized exponential distribution,” Aust NZJ Stat, vol. 41, pp. 173–188, 1999.
[14]	R. D. Gupta and D. Kundu, “Generalized exponential distribution: different method of estimations,” Journal of Statistical Computation and Simulation, vol. 69, no. 4, pp. 315–337, 2001b.
[15]	R. D. Gupta and D. Kundu, “Closeness of gamma and generalized exponential distribution,” Communications in statistics-theory and methods, vol. 32, no. 4, pp. 705–721, 2003a.
[16]	R. D. Gupta and D. Kundu, “Discriminating between weibull and generalized exponential distributions,” Computational statistics & data analysis, vol. 43, no. 2, pp. 179–196, 2003b.
[17]	R. D. Gupta and D. Kundu, “Discriminating between gamma and generalized exponential distributions,” Journal of Statistical Computation & Simulation, vol. 74, no. 2, pp. 107–121, 2004.
[18]	R. D. Gupta and D. Kundu, “On the comparison of fisher information of the weibull and ge distributions,” Journal of Statistical Planning and Inference, vol. 136, no. 9, pp. 3130–3144, 2006.
[19]	R. D. Gupta and D. Kundu, “Generalized exponential distribution: Existing results and some recent developments,” Journal of Statistical Planning and Inference, vol. 137, no. 11, pp. 3537–3547, 2007.
[20]	R. D. Gupta and D. Kundu, “Generalized exponential distribution: Bayesian inference,” Computational Statistics and Data Analysis, vol. 52, no. 4, pp. 1873–1883, 2008.
[21]	D. Kundu, R. D. Gupta, and A. Manglick, “Discriminating between the log-normal and generalized exponential distributions,” Journal of Statistical Planning and Inference, vol. 127, no. 1-2, pp. 213–227, 2005.
[22]	A. K. Dey and D. Kundu, “Discriminating among the log-normal, weibull, and generalized exponential distributions,” IEEE Transactions on reliability, vol. 58, no. 3, pp. 416–424, 2009.
[23]	A. Asgharzadeh and R. Rezaei, “The generalized exponential distribution as a lifetime model under different loss functions,” Data science journal, vol. 8, pp. 217–225, 2009.
[24]	R. Pakyari, “Discriminating between generalized exponential, geometric extreme exponential and weibull distributions,” Journal of statistical computation and simulation, vol. 80, no. 12, pp. 1403–1412, 2010.
[25]	M. Y. Danish and M. Aslam, “Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions,” Journal of Applied Statistics, vol. 40, no. 5, pp. 1106–1119, 2013.
[26]	M. Mohie El-Din, M. Amein, A. Shafay, and S. Mohamed, “Estimation of generalized exponential distribution based on an adaptive progressively type-ii censored sample,” Journal of Statistical Computation and Simulation, vol. 87, no. 7, pp. 1292–1304, 2017.
[27]	D. Kundu and R. D. Gupta, “Bivariate generalized exponential distribution,” Journal of multivariate analysis, vol. 100, no. 4, pp. 581–593, 2009.
[28]	F. Merovci, “Transmuted exponentiated exponential distribution,” Mathematical Sciences and Applications ENotes, vol. 1, no. 2, pp. 112–122, 2013.
[29]	V. Nekoukhou and D. Kundu, “Bivariate discrete generalized exponential distribution,” Statistics, vol. 51, no. 5, pp. 1143–1158, 2017.
[30]	S. S. Maiti and S. Pramanik, “Odds generalized exponential-exponential distribution,” Journal of data science, vol. 13, no. 4, pp. 733–753, 2015.
[31]	M. K. A. Elaal and R. S. Jarwan, “Inference of bivariate generalized exponential distribution based on copula functions,” Applied Mathematical Sciences, vol. 11, no. 24, pp. 1155–1186, 2017.
[32]	B. Abdulwasiu and G. Oyeyemi, “On development of four parameters exponentiated generalized exponential distribution,” Pakistan Journal of Statistics, vol. 34, no. 4, 2018.
[33]	G. M. Cordeiro, E. M. Ortega, and T. G. Ramires, “A new generalized weibull family of distributions: mathematical properties and applications,” Journal of Statistical Distributions and Applications, vol. 2, no. 1, p. 13, 2015.
[34]	S. Nadarajah, G. M. Cordeiro, and E. M. Ortega, “The zografos–balakrishnan-g family of distributions: mathematical properties and applications,” Communications in Statistics-Theory and Methods, vol. 44, no. 1, pp. 186–215, 2015.
[35]	F. Galton, Inquiries into human faculty and its development. Macmillan, 1883.
[36]	J. Moors, “A quantile alternative for kurtosis,” Journal of the Royal Statistical Society: Series D (The Statistician), vol. 37, no. 1, pp. 25–32, 1988.
[37]	M. V. Aarset, “How to identify a bathtub hazard rate,” IEEE Transactions on Reliability, vol. 36, no. 1, pp. 106–108, 1987.

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APA Style

Umar Usman, Suleiman Shamsuddeen, Bello Magaji Arkilla, Yakubu Aliyu. (2020). Inferences on the Weibull Exponentiated Exponential Distribution and Applications. International Journal of Statistical Distributions and Applications, 6(1), 10-22. https://doi.org/10.11648/j.ijsd.20200601.12

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ACS Style

Umar Usman; Suleiman Shamsuddeen; Bello Magaji Arkilla; Yakubu Aliyu. Inferences on the Weibull Exponentiated Exponential Distribution and Applications. Int. J. Stat. Distrib. Appl. 2020, 6(1), 10-22. doi: 10.11648/j.ijsd.20200601.12

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AMA Style

Umar Usman, Suleiman Shamsuddeen, Bello Magaji Arkilla, Yakubu Aliyu. Inferences on the Weibull Exponentiated Exponential Distribution and Applications. Int J Stat Distrib Appl. 2020;6(1):10-22. doi: 10.11648/j.ijsd.20200601.12

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@article{10.11648/j.ijsd.20200601.12,
  author = {Umar Usman and Suleiman Shamsuddeen and Bello Magaji Arkilla and Yakubu Aliyu},
  title = {Inferences on the Weibull Exponentiated Exponential Distribution and Applications},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {6},
  number = {1},
  pages = {10-22},
  doi = {10.11648/j.ijsd.20200601.12},
  url = {https://doi.org/10.11648/j.ijsd.20200601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20200601.12},
  abstract = {In this article, an alternative method of defining the probability density function of GeneralizedWeibull-exponential distributions is proposed. Based on the method, the distribution can also be calledWeibull exponentiated exponential distribution. This distribution includes the exponential, Weibull and exponentiated exponential distributions as special cases. Comprehensive mathematical treatment of the distribution is provided. The quantile function, mode, characteristic function, moment generating function among other mathematical properties of the distribution were derived. The parameters of the distribution were estimated by applying the Maximum Likelihood Procedure.The elements of the Fisher Information Matrix is also provided. Finally, a data set is fitted to the model and its sub-models. It is observed that the new distribution is more flexible and can be used quiet effectively in analysing real life data in place of exponential, Weibull and exponentiated exponential distributions.},
 year = {2020}
}

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TY  - JOUR
T1  - Inferences on the Weibull Exponentiated Exponential Distribution and Applications
AU  - Umar Usman
AU  - Suleiman Shamsuddeen
AU  - Bello Magaji Arkilla
AU  - Yakubu Aliyu
Y1  - 2020/07/15
PY  - 2020
N1  - https://doi.org/10.11648/j.ijsd.20200601.12
DO  - 10.11648/j.ijsd.20200601.12
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  - 10
EP  - 22
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20200601.12
AB  - In this article, an alternative method of defining the probability density function of GeneralizedWeibull-exponential distributions is proposed. Based on the method, the distribution can also be calledWeibull exponentiated exponential distribution. This distribution includes the exponential, Weibull and exponentiated exponential distributions as special cases. Comprehensive mathematical treatment of the distribution is provided. The quantile function, mode, characteristic function, moment generating function among other mathematical properties of the distribution were derived. The parameters of the distribution were estimated by applying the Maximum Likelihood Procedure.The elements of the Fisher Information Matrix is also provided. Finally, a data set is fitted to the model and its sub-models. It is observed that the new distribution is more flexible and can be used quiet effectively in analysing real life data in place of exponential, Weibull and exponentiated exponential distributions.
VL  - 6
IS  - 1
ER  -

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Author Information

Umar Usman

Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria
Suleiman Shamsuddeen

Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria
Bello Magaji Arkilla

Department of Community Health, Usmanu Danfodiyo University, Sokoto, Nigeria
Yakubu Aliyu

Department of Statistics, Ahmadu Bello University, Zaria, Nigeria

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Plain Text BibTeX RIS

APA Style

Umar Usman, Suleiman Shamsuddeen, Bello Magaji Arkilla, Yakubu Aliyu. (2020). Inferences on the Weibull Exponentiated Exponential Distribution and Applications. International Journal of Statistical Distributions and Applications, 6(1), 10-22. https://doi.org/10.11648/j.ijsd.20200601.12

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ACS Style

Umar Usman; Suleiman Shamsuddeen; Bello Magaji Arkilla; Yakubu Aliyu. Inferences on the Weibull Exponentiated Exponential Distribution and Applications. Int. J. Stat. Distrib. Appl. 2020, 6(1), 10-22. doi: 10.11648/j.ijsd.20200601.12

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AMA Style

Umar Usman, Suleiman Shamsuddeen, Bello Magaji Arkilla, Yakubu Aliyu. Inferences on the Weibull Exponentiated Exponential Distribution and Applications. Int J Stat Distrib Appl. 2020;6(1):10-22. doi: 10.11648/j.ijsd.20200601.12

Copy | Download

@article{10.11648/j.ijsd.20200601.12,
  author = {Umar Usman and Suleiman Shamsuddeen and Bello Magaji Arkilla and Yakubu Aliyu},
  title = {Inferences on the Weibull Exponentiated Exponential Distribution and Applications},
  journal = {International Journal of Statistical Distributions and Applications},
  volume = {6},
  number = {1},
  pages = {10-22},
  doi = {10.11648/j.ijsd.20200601.12},
  url = {https://doi.org/10.11648/j.ijsd.20200601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20200601.12},
  abstract = {In this article, an alternative method of defining the probability density function of GeneralizedWeibull-exponential distributions is proposed. Based on the method, the distribution can also be calledWeibull exponentiated exponential distribution. This distribution includes the exponential, Weibull and exponentiated exponential distributions as special cases. Comprehensive mathematical treatment of the distribution is provided. The quantile function, mode, characteristic function, moment generating function among other mathematical properties of the distribution were derived. The parameters of the distribution were estimated by applying the Maximum Likelihood Procedure.The elements of the Fisher Information Matrix is also provided. Finally, a data set is fitted to the model and its sub-models. It is observed that the new distribution is more flexible and can be used quiet effectively in analysing real life data in place of exponential, Weibull and exponentiated exponential distributions.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Inferences on the Weibull Exponentiated Exponential Distribution and Applications
AU  - Umar Usman
AU  - Suleiman Shamsuddeen
AU  - Bello Magaji Arkilla
AU  - Yakubu Aliyu
Y1  - 2020/07/15
PY  - 2020
N1  - https://doi.org/10.11648/j.ijsd.20200601.12
DO  - 10.11648/j.ijsd.20200601.12
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  - 10
EP  - 22
PB  - Science Publishing Group
SN  - 2472-3509
UR  - https://doi.org/10.11648/j.ijsd.20200601.12
AB  - In this article, an alternative method of defining the probability density function of GeneralizedWeibull-exponential distributions is proposed. Based on the method, the distribution can also be calledWeibull exponentiated exponential distribution. This distribution includes the exponential, Weibull and exponentiated exponential distributions as special cases. Comprehensive mathematical treatment of the distribution is provided. The quantile function, mode, characteristic function, moment generating function among other mathematical properties of the distribution were derived. The parameters of the distribution were estimated by applying the Maximum Likelihood Procedure.The elements of the Fisher Information Matrix is also provided. Finally, a data set is fitted to the model and its sub-models. It is observed that the new distribution is more flexible and can be used quiet effectively in analysing real life data in place of exponential, Weibull and exponentiated exponential distributions.
VL  - 6
IS  - 1
ER  -

Copy | Download