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Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.37 no.1 Bogotá Jan./June 2014

http://dx.doi.org/10.15446/rce.v37n1.44363 

http://dx.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


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[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}
}