Published

2014-01-01

The Beta-Gompertz Distribution

La distribución Beta-Gompertz

DOI:

https://doi.org/10.15446/rce.v37n1.44363

Keywords:

Beta generator, Gompertz distribution, Maximum likelihood estimation. (en)
distribución de Gompertz, estimación máximo verosímil, función Beta (es)

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Authors

  • Ali Akbar Jafari Yazd University
  • Saeid Tahmasebi Persian Gulf University
  • Morad Alizadeh Ferdowsi University of Mashhad

In this paper, we introduce a new four-parameter generalized version of the Gompertz model which is called Beta-Gompertz (BG) distribution. It includes some well-known lifetime distributions such as Beta-exponential and generalized Gompertz distributions as special sub-models. This new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a decreasing, increasing, and bathtub-shaped failure rate function depending on its parameters. Some mathematical properties of the new distribution, such as closed-form expressions for the density, cumulative distribution, hazard rate function, the kth order moment, moment generating function, Shannon entropy, and the quantile measure are provided. We discuss maximum likelihood estimation of the BG parameters from one observed sample and derive the observed Fisher’s information matrix. A simulation study is performed in order to investigate the properties of the proposed estimator. At the end, in order to show the BG distribution flexibility, an application using a real data set is presented.

En este artículo, se introduce una versión generalizada en cuatro parámetros de la distribución de Gompertz denominada como la distribución Beta-Gompertz (BG). Esta incluye algunas distribuciones de duración de vida bien conocidas como la Beta exponencial y distribuciones Gompertz generalizadas como casos especiales. Esta nueva distribución es flexible y puede ser usada de manera efectiva en datos de sobrevida y problemas de confiabilidad. Su función de tasa de falla puede ser decreciente, creciente o en forma de bañera dependiendo de sus parámetros. Algunas propiedades matemáticas de la distribución como expresiones en forma cerrada para la densidad, función de distribución, función de riesgo, momentos k-ésimos, función generadora de momentos, entropía de Shannon y cuantiles son presentados. Se discute la estimación máximo verosímil de los parámetros desconocidos del nuevo modelo para la muestra completa y se obtiene una expresión para la matriz de información. Con el fin de mostrar la flexibilidad de esta distribución, se presenta una aplicación con datos reales. Al final, un estudio de simulación es desarrollado.

https://doi.org/10.15446/rce.v37n1.44363

The Beta-Gompertz Distribution

La distribución Beta-Gompertz

ALI AKBAR JAFARI1, SAEID TAHMASEBI2, MORAD ALIZADEH3

1Yazd University, Department of Statistics, Yazd, Iran. Professor. Email: aajafari@yazd.ac.ir
2Persian Gulf University, Department of Statistics, Bushehr, Iran. Professor. Email: tahmasebi@pgu.ac.ir
3Ferdowsi University of Mashhad, Department of Statistics, Mashhad, Iran. Ph.D Student. Email: moradalizadeh78@gmail.com

Abstract

In this paper, we introduce a new four-parameter generalized version of the Gompertz model which is called Beta-Gompertz (BG) distribution. It includes some well-known lifetime distributions such as Beta-exponential and generalized Gompertz distributions as special sub-models. This new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a decreasing, increasing, and bathtub-shaped failure rate function depending on its parameters. Some mathematical properties of the new distribution, such as closed-form expressions for the density, cumulative distribution, hazard rate function, the kth order moment, moment generating function, Shannon entropy, and the quantile measure are provided. We discuss maximum likelihood estimation of the BG parameters from one observed sample and derive the observed Fishers information matrix. A simulation study is performed in order to investigate the properties of the proposed estimator. At the end, in order to show the BG distribution flexibility, an application using a real data set is presented.

Key words: Beta generator, Gompertz distribution, Maximum likelihood estimation.

Resumen

En este artículo, se introduce una versión generalizada en cuatro parámetros de la distribución de Gompertz denominada como la distribución Beta-Gompertz (BG). Esta incluye algunas distribuciones de duración de vida bien conocidas como la Beta exponencial y distribuciones Gompertz generalizadas como casos especiales. Esta nueva distribución es flexible y puede ser usada de manera efectiva en datos de sobrevida y problemas de confiabilidad. Su función de tasa de falla puede ser decreciente, creciente o en forma de bañera dependiendo de sus parámetros. Algunas propiedades matemáticas de la distribución como expresiones en forma cerrada para la densidad, función de distribución, función de riesgo, momentos k-ésimos, función generadora de momentos, entropía de Shannon y cuantiles son presentados. Se discute la estimación máximo verosímil de los parámetros desconocidos del nuevo modelo para la muestra completa y se obtiene una expresión para la matriz de información. Con el fin de mostrar la flexibilidad de esta distribución, se presenta una aplicación con datos reales. Al final, un estudio de simulación es desarrollado.

Palabras clave: distribución de Gompertz, estimación máximo verosímil, función Beta.

Texto completo disponible en PDF

References

1. A. Akinsete,, F. Famoye, & C. Lee, (2008), 'The Beta-Pareto distribution', Statistics 42(6), 547-563.

2. A. C. Bemmaor, & N. Glady, (2012), 'Modeling purchasing behavior with sudden ''death'': A flexible customer lifetime model', Management Science 58(5), 1012-1021.

3. A. C. Economos, (1982), 'Rate of aging, rate of dying and the mechanism of mortality', Archives of Gerontology and Geriatrics 1(1), 46-51.

4. Alshamrani El-Gohary, & A. N. Al-Otaibi, (2013), 'The generalized Gompertz distribution', Applied Mathematical Modelling 37(1-2), 13-24.

5. C. Shannon, (1948), 'A mathematical theory of communication', Bell System Technical Journal 27, 379-432.

6. F. Famoye,, C. Lee, & O. Olumolade, (2005), 'The Beta-Weibull distribution', Journal of Statistical Theory and Applications 4(2), 121-136.

7. G. M. Cordeiro, & S. Nadarajah, (2011), 'Closed-form expressions for moments of a class of Beta generalized distributions', Brazilian Journal of Probability and Statistics 25(1), 14-33.

8. G. O., Ortega, E. M. Silva, & G. M. Cordeiro, (2010), 'The Beta modified Weibull distribution', Lifetime Data Analysis 16(3), 409-430.

9. I. S. Gradshteyn, & I. M. Ryzhik, (2007), Table of Integrals, Series, and Products, 7 edn, Academic Press, New York.

10. J. F. Kenney, & E. Keeping, (1962), Mathematics of Statistics, Van Nostrand.

11. J. J. A. Moors, (1988), 'A quantile alternative for kurtosis', Journal of the Royal Statistical Society. Series D (The Statistician) 37(1), 25-32.

12. K. Brown, & W. Forbes, (1974), 'A mathematical model of aging processes', Journal of Gerontology 29(1), 46-51.

13. K. Ohishi,, H. Okamura, & T. Dohi, (2009), 'Gompertz software reliability model: Estimation algorithm and empirical validation', Journal of Systems and Software 82(3), 535-543.

14. M. V. Aarset, (1987), 'How to identify a bathtub hazard rate', IEEE Transactions on Reliability 36(1), 106-108.

15. N. Eugene,, C. Lee, & F. Famoye, (2002), 'Beta-normal distribution and its applications', Communications in Statistics - Theory and Methods 31(4), 497-512.

16. N. L. Johnson,, S. Kotz, & N. Balakrishnan, (1995), Continuous Univariate Distributions, Vol. 2, 2 edn, John Wiley & Sons, New York.

17. R. C. Gupta, & R. D. Gupta, (2007), 'Proportional reversed hazard rate model and its applications', Journal of Statistical Planning and Inference 137(11), 3525-3536.

18. R. D. Gupta, & D. Kundu, (1999), 'Generalized exponential distributions', Australian & New Zealand Journal of Statistics 41(2), 173-188.

19. R. J., Rãgo, L. C., Cordeiro, G. M. Cintra, & A. D. C. Nascimento, (2012), 'Beta generalized normal distribution with an application for SAR image processing', Statistics: A Journal of Theoretical and Applied Statistics, 1-16. DOI:10.1080/02331888.2012.748776.

20. S. Nadarajah, & S. Kotz, (2004), 'The Beta Gumbel distribution', Mathematical Problems in Engineering 4, 323-332.

21. S. Nadarajah, & S. Kotz, (2006), 'The Beta exponential distribution', Reliability Engineering & System Safety 91(6), 689-697.

22. W. Willemse, & H. Koppelaar, (2000), 'Knowledge elicitation of Gompertz' law of mortality', Scandinavian Actuarial Journal 2, 168-179.

[Recibido en octubre de 2013. Aceptado en marzo de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv37n1a10,
    AUTHOR  = {Jafari, Ali Akbar and Tahmasebi, Saeid and Alizadeh, Morad},
    TITLE   = {{The Beta-Gompertz Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2014},
    volume  = {37},
    number  = {1},
    pages   = {141-158}
}

References

Aarset, M. V. (1987), ‘How to identify a bathtub hazard rate’, IEEE Transactions on Reliability 36(1), 106–108.

Akinsete, A., Famoye, F. & Lee, C. (2008), ‘The Beta-Pareto distribution’, Statistics 42(6), 547–563.

Bemmaor, A. C. & Glady, N. (2012), ‘Modeling purchasing behavior with sudden “death”: A flexible customer lifetime model’, Management Science 58(5), 1012–1021.

Brown, K. & Forbes, W. (1974), ‘A mathematical model of aging processes’, Journal of Gerontology 29(1), 46–51.

Cintra, R. J., R. L. C. C. G. M. & Nascimento, A. D. C. (2012), ‘Beta generalized normal distribution with an application for SAR image processing’, Statistics: A Journal of Theoretical and Applied Statistics pp. 1–16. DOI:10.1080/02331888.2012.748776.

Cordeiro, G. M. & Nadarajah, S. (2011), ‘Closed-form expressions for moments of a class of Beta generalized distributions’, Brazilian Journal of Probability and Statistics 25(1), 14–33.

Economos, A. C. (1982), ‘Rate of aging, rate of dying and the mechanism of mortality’, Archives of Gerontology and Geriatrics 1(1), 46–51.

El-Gohary, A. & Al-Otaibi, A. N. (2013), ‘The generalized Gompertz distribution’, Applied Mathematical Modelling 37(1-2), 13–24.

Eugene, N., Lee, C. & Famoye, F. (2002), ‘Beta-normal distribution and its applications’, Communications in Statistics - Theory and Methods 31(4), 497–512.

Famoye, F., Lee, C. & Olumolade, O. (2005), ‘The Beta-Weibull distribution’, Journal of Statistical Theory and Applications 4(2), 121–136.

Gradshteyn, I. S. & Ryzhik, I. M. (2007), Table of Integrals, Series, and Products, 7 edn, Academic Press, New York.

Gupta, R. C. & Gupta, R. D. (2007), ‘Proportional reversed hazard rate model and its applications’, Journal of Statistical Planning and Inference 137(11), 3525–3536.

Gupta, R. D. & Kundu, D. (1999), ‘Generalized exponential distributions’, Australian & New Zealand Journal of Statistics 41(2), 173–188.

Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995), Continuous Univariate Distributions, Vol. 2, 2 edn, John Wiley & Sons, New York.

Kenney, J. F. & Keeping, E. (1962), Mathematics of Statistics, Van Nostrand.

Moors, J. J. A. (1988), ‘A quantile alternative for kurtosis’, Journal of the Royal Statistical Society. Series D (The Statistician) 37(1), 25–32.

Nadarajah, S. & Kotz, S. (2004), ‘The Beta Gumbel distribution’, Mathematical Problems in Engineering 4, 323–332.

Nadarajah, S. & Kotz, S. (2006), ‘The Beta exponential distribution’, Reliability Engineering & System Safety 91(6), 689–697.

Ohishi, K., Okamura, H. & Dohi, T. (2009), ‘Gompertz software reliability model: Estimation algorithm and empirical validation’, Journal of Systems and Software 82(3), 535–543.

Shannon, C. (1948), ‘A mathematical theory of communication’, Bell System Technical Journal 27, 379–432.

Silva, G. O., O. E. M. & Cordeiro, G. M. (2010), ‘The Beta modified Weibull distribution’, Lifetime Data Analysis 16(3), 409–430.

Willemse, W. & Koppelaar, H. (2000), ‘Knowledge elicitation of Gompertz’ law of mortality’, Scandinavian Actuarial Journal 2, 168–179.

How to Cite

APA

Jafari, A. A., Tahmasebi, S. and Alizadeh, M. (2014). The Beta-Gompertz Distribution. Revista Colombiana de Estadística, 37(1), 141–158. https://doi.org/10.15446/rce.v37n1.44363

ACM

[1]
Jafari, A.A., Tahmasebi, S. and Alizadeh, M. 2014. The Beta-Gompertz Distribution. Revista Colombiana de Estadística. 37, 1 (Jan. 2014), 141–158. DOI:https://doi.org/10.15446/rce.v37n1.44363.

ACS

(1)
Jafari, A. A.; Tahmasebi, S.; Alizadeh, M. The Beta-Gompertz Distribution. Rev. colomb. estad. 2014, 37, 141-158.

ABNT

JAFARI, A. A.; TAHMASEBI, S.; ALIZADEH, M. The Beta-Gompertz Distribution. Revista Colombiana de Estadística, [S. l.], v. 37, n. 1, p. 141–158, 2014. DOI: 10.15446/rce.v37n1.44363. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44363. Acesso em: 28 mar. 2025.

Chicago

Jafari, Ali Akbar, Saeid Tahmasebi, and Morad Alizadeh. 2014. “The Beta-Gompertz Distribution”. Revista Colombiana De Estadística 37 (1):141-58. https://doi.org/10.15446/rce.v37n1.44363.

Harvard

Jafari, A. A., Tahmasebi, S. and Alizadeh, M. (2014) “The Beta-Gompertz Distribution”, Revista Colombiana de Estadística, 37(1), pp. 141–158. doi: 10.15446/rce.v37n1.44363.

IEEE

[1]
A. A. Jafari, S. Tahmasebi, and M. Alizadeh, “The Beta-Gompertz Distribution”, Rev. colomb. estad., vol. 37, no. 1, pp. 141–158, Jan. 2014.

MLA

Jafari, A. A., S. Tahmasebi, and M. Alizadeh. “The Beta-Gompertz Distribution”. Revista Colombiana de Estadística, vol. 37, no. 1, Jan. 2014, pp. 141-58, doi:10.15446/rce.v37n1.44363.

Turabian

Jafari, Ali Akbar, Saeid Tahmasebi, and Morad Alizadeh. “The Beta-Gompertz Distribution”. Revista Colombiana de Estadística 37, no. 1 (January 1, 2014): 141–158. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44363.

Vancouver

1.
Jafari AA, Tahmasebi S, Alizadeh M. The Beta-Gompertz Distribution. Rev. colomb. estad. [Internet]. 2014 Jan. 1 [cited 2025 Mar. 28];37(1):141-58. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44363

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