Four generalized Weibull distributions: similar properties and applications

Authors

DOI:

https://doi.org/10.5902/2179460X40100

Keywords:

Beta Weibul, Gamma-Weibull, Kuramaswamy-Weibull, Marshall-Olkin-Weibull

Abstract

We derive a common linear representation for the densities of four generalizations of the two-parameter Weibull distribution in terms of Weibull densities. The four generalized Weibull distributions briefly studied are: the Marshall-Olkin-Weibull, beta-Weibull, gamma-Weibull and Kumaraswamy-Weibull distributions. We demonstrate that several mathematical properties of these generalizations can be obtained simultaneously from those of the Weibull properties. We present two applications to real data sets by comparing these generalized distributions. It is hoped that this paper encourage developments of further generalizations of the Weibull based on the same linear representation.

Downloads

Download data is not yet available.

Author Biographies

Edwin Moises Marcos Ortega, Universidade de São Paulo, Piracicaba, SP

Estatístico formado pela Universidad Nacional Mayor de San Marcos, Mestre em Estatística pela Universidade Estadual de Campinas, Doutor em Estatística pela Universidade de São Paulo, livre docente em e Pós-doutor em Estatística na Universidade Federal de Pernambuco

Fábio Prataviera, Universidade de São Paulo, Piracicaba, SP

Bacharel em Estatística pela Universidade Federal de São Carlos, Mestre em Ciências, Área: Estatística e Experimentação Agronômica pela Escola Superior de Agricultura Luiz de Queiroz - ESALQ/USP e Doutor em Ciências, Área: Estatística e Experimentação Agronômica, pela Escola Superior de Agricultura Luiz de Queiroz - ESALQ/USP 

Gauss Moutinho Cordeiro, Universidade Federal de Pernambuco, Recife, PE

Bacharel em Matemática pela Universidade Católica de Pernambuco, graduação em Engenharia Civil pela Universidade Federal de Pernambuco, Mestre em Engenharia de Produção pela Universidade Federal do Rio de Janeiro e Doutor em Estatística pelo Imperial College, University of London. Pós-doutor na University of London e no Instituto de Matemática Pura e Aplicada

References

Alexander, C., Cordeiro, G. M., Ortega, E. M. M. and Sarabia, J. M. (2012). Generalized beta-generated distributions. Computational Statistics and Data Analysis, 56, 1880-1897.

Barreto-Souza, W., Santos, A. and Cordeiro, G. M. (2010). The beta generalized exponential distribution. Journal of Statistical Computation and Simulation, 80, 159-172.

Bebbington, M., Lai, C.D. and Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering and System Safety, 92, 719-726.

Carrasco, J. M. F., Ortega, E. M. M. and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics and Data Analysis, 53, 450-462.

Castellares, F. and Lemonte, A. (2015). A new generalized Weibull distribution generated by gamma random variables. Journal of the Egyptian Mathematical Society, 23, 382-390.

Chhikara, R. S. and Folks, J. L. (1977). The Inverse Gaussian Distribution as a Lifetime Model. Technometrics, 19, 461.

Cordeiro, G. M. and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81, 883-898.

Cordeiro, G. M., Ortega, E. M. M. and Silva, G. (2011b). The exponentiated generalized gamma distribution with application to lifetime data. Journal of Statistical Computation and Simulation, 81, 827-842.

Cordeiro, G. M., Ortega, E. M. M. and Silva, G. (2011c). The beta extended Weibull family. Journal of Probability and Statistical Science, 10, 15-40.

Cordeiro, G. M., Lemonte, A. and Ortega, E. M. M. (2014). The Marshall–Olkin Family of Distributions: Mathematical Properties and New Models. Journal of Statistical Theory and Practice, 8, 343-366.

Cordeiro, G. M., Lemonte, A. and Ortega, E. M. M. (2014). The Marshall–Olkin family of distributions: Mathematical properties and new models. Journal of Statistical Theory and Practice, 8, 343-366.

Eugene, M, Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communication in Statistics - Theory Methods, 31, 497-512.

Gera, A. W. (1997). The modified exponentiated-Weibull distribution for lifetime modeling. In: 1997 Proceedings of the Annual Reliability and Maintainability Symposium, pp. 149-152.

Kumagai, S. and Matsunaga, I. (1995). Changes in the distribution of short-term exposure concentration with different averaging times. American Industrial Hygiene Association Journal, 56, 24- 31.

Lai, C. D., Xie, M. and Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52, 33-37.

Marshall, A. W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, 641- 652.

Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42, 299-302.

Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: some properties and a flood data application. Communications in Statistics—Theory and Methods, 25, 3059-3083.

Nadarajah S. and Kotz S. (2005). On some recent modifications of Weibull distribution. IEEE Transactions on Reliability, 54, 561-562.

Nadarajah, S., Cordeiro, G. M., and Ortega, E. M. M. (2012). General results for the Kumaraswamy G distribution. Journal of Statistical Computation and Simulation, 82, 951-979.

Nikulin, M. and Haghighi, F. (2009). On the power generalized Weibull family. Metron, 67, 75-86.

Pascoa, M. A. P., Ortega, E. M. M. and Cordeiro, G. M. (2011). The Kumaraswamygeneralized gamma distribution with application in survival analysis. Statistical Methodology, 8, 411-433.

Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series, volumes 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam.

Silva, G. O., Ortega, E. M. M. and Cordeiro, G. M. (2010). The beta modified Weibull distribution. Lifetime Data Analysis, 16, 409-430.

Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics, 33, 1187-1192.

Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma generated distributions and associated inference. Statistical Methodology, 6, 344-362.

Downloads

Published

2020-09-03

How to Cite

Ortega, E. M. M., Prataviera, F., & Cordeiro, G. M. (2020). Four generalized Weibull distributions: similar properties and applications. Ciência E Natura, 42, e10. https://doi.org/10.5902/2179460X40100

Issue

Vol. 42 (2020): 40 years special anniversary edition

Section

40 YEARS - Anniversary Special Edition

License

To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.

 

Ethical Guidelines for Journal Publication

The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles. 

Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.

In particular, 

Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.

Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.

Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.

Most read articles by the same author(s)