Published

2014-01-01

The Poisson-Lomax Distribution

Distribución Poisson-Lomax

DOI:

https://doi.org/10.15446/rce.v37n1.44369

Keywords:

Asymptotic variance-covariance matrix, Compounding, Lifetime distributions, Lomax distribution, Poisson distribution, Maximum likelihood estimation. (en)
mezclas, distribuciones de sobrevida, distribució n Lomax, distribución Poisson, estomación máximo-verosímil (es)

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Authors

  • Bander Al-Zahrani King Abdulaziz University
  • Hanaa Sagor King Abdulaziz University

In this paper we propose a new three-parameter lifetime distribution
with upside-down bathtub shaped failure rate. The distribution is a compound distribution of the zero-truncated Poisson and the Lomax distributions (PLD). The density function, shape of the hazard rate function, a general expansion for moments, the density of the rth order statistic, and the mean and median deviations of the PLD are derived and studied in detail. The maximum likelihood estimators of the unknown parameters are
obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, a real data set is analyzed to show the potential of the new proposed distribution.

En este artículo se propone una nueva distribución de sobrevida de tres parámetros con tasa fallo en forma de bañera. La distribución es una mezclade la Poisson truncada y la distribución Lomax. La  función de densidad, la función de riesgo, una expansión general de los momentos, la densidad del r-ésimo estadístico de orden, y la media así como su desviación estándar son derivadas y estudiadas en detalle. Los estimadores de máximo verosímiles de los parámetros desconocidos son obtenidos. Los intervalos de confianza asintóticas se obtienen según la matriz de varianzas y covarianzas asintótica. Finalmente, un conjunto de datos reales es analizado para construir el potencial de la nueva distribución propuesta.

https://doi.org/10.15446/rce.v37n1.44369

The Poisson-Lomax Distribution

Distribución Poisson-Lomax

BANDER AL-ZAHRANI1, HANAA SAGOR2

1King Abdulaziz University, Department of Statistics, Jeddah, Saudi Arabia. Professor. Email: bmalzahrani@kau.edu.sa
2King Abdulaziz University, Department of Statistics, Jeddah, Saudi Arabia. Ph.D student. Email: hsagor123@gmail.com

Abstract

In this paper we propose a new three-parameter lifetime distribution with upside-down bathtub shaped failure rate. The distribution is a compound distribution of the zero-truncated Poisson and the Lomax distributions (PLD). The density function, shape of the hazard rate function, a general expansion for moments, the density of the rth order statistic, and the mean and median deviations of the PLD are derived and studied in detail. The maximum likelihood estimators of the unknown parameters are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, a real data set is analyzed to show the potential of the new proposed distribution.

Key words: Asymptotic variance-covariance matrix, Compounding, Lifetime distributions, Lomax distribution, Poisson distribution, Maximum likelihood estimation.

Resumen

En este artículo se propone una nueva distribución de sobrevida de tres parámetros con tasa fallo en forma de bañera. La distribución es una mezcla de la Poisson truncada y la distribución Lomax. La función de densidad, la función de riesgo, una expansión general de los momentos, la densidad del r-ésimo estadístico de orden, y la media así como su desviación estándar son derivadas y estudiadas en detalle. Los estimadores de máximo verosímiles de los parámetros desconocidos son obtenidos. Los intervalos de confianza asintóticas se obtienen según la matriz de varianzas y covarianzas asintótica. Finalmente, un conjunto de datos reales es analizado para construir el potencial de la nueva distribución propuesta.

Palabras clave: mezclas, distribuciones de sobrevida, distribució n Lomax, distribución Poisson, estomación máximo-verosímil.

Texto completo disponible en PDF

References

1. A. W. Marshall, & I. Olkin, (1997), 'A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families', Biometrika 84(3), 641-652.

2. B. Al-Zahrani, (2012), 'Goodness-of-Fit for the Topp-Leone Distribution with Unknown Parameters', Applied Mathematical Sciences 6(128), 6355-6363.

3. B. C. Arnold,, N. Balakrishnan, & H. H. N. Nagaraja, (1992), A First Course in Order Statistics, John Wiley & Sons, New York.

4. C. Kus, (2007), 'A new lifetime distribution', Computational Statistics & Data Analysis 51(9), 4497-4509.

5. E. T. Lee, & J. W. Wang, (2003), Statistical Methods for Survival Data Analysis, 3 edn, John Wiley, New York.

6. H.A. David, & H. N. Nagaraja, (2003), Order Statistics, John Wiley & Sons, Hoboken, New Jersey.

7. Jr. R.G. Miller, (1981), Survival Analysis, John Wiley, New York.

8. M. E. Ghitany,, E. K. Al-Hussaini, & R. A. Al-Jarallah, (2005), 'Marshall-Olkin extended Weibull distribution and its application to censored data', Journal of Applied Statistics 32(10), 1025-1034.

9. M. E. Ghitany,, F. A. Al-Awadhi, & L. A. Alkhalfan, (2007), 'Marshall-Olkin extended Lomax distribution and its application to censored data', Communications in Statistics-Theory and Methods 36(10), 1855-1866.

10. S. A. Al-Awadhi, & M. E. Ghitany, (2001), 'Statistical properties of Poisson-Lomax distribution and its application to repeated accidents data', Journal of Applied Statistical Sciences 10(4), 365-372.

11. T. Alice, & K. K. Jose, (2003), 'Marshall-Olkin Pareto processes', Far East Journal of Theoretical Statistics 2(9), 117-132.

12. V. G. Cancho,, F. Louzada-Neto, & G. D. Barriga, (2011), 'The Poisson-exponential lifetime distribution', Computational Statistics & Data Analysis 55(1), 677-686.

[Recibido en noviembre de 2013. Aceptado en abril de 2014]

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

@ARTICLE{RCEv37n1a15,
    AUTHOR  = {Al-Zahrani, Bander and Sagor, Hanaa},
    TITLE   = {{The Poisson-Lomax Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2014},
    volume  = {37},
    number  = {1},
    pages   = {225-245}
}

References

Al-Awadhi, S. A. & Ghitany, M. E. (2001), ‘Statistical properties of Poisson-Lomax distribution and its application to repeated accidents data’, Journal of Applied Statistical Sciences 10(4), 365–372.

Al-Zahrani, B. (2012), ‘Goodness-of-fit for the Topp-Leone distribution with unknown parameters’, Applied Mathematical Sciences 6(128), 6355–6363.

Alice, T. & Jose, K. K. (2003), ‘Marshall-Olkin Pareto processes’, Far East Journal of Theoretical Statistics 2(9), 117–132.

Arnold, B. C., Balakrishnan, N. & Nagaraja, H. H. N. (1992), A First Course in Order Statistics, John Wiley & Sons, New York.

Cancho, V. G., Louzada-Neto, F. & Barriga, G. D. (2011), ‘The Poisson-exponential lifetime distribution’, Computational Statistics & Data Analysis 55(1), 677–686.

David, H. & Nagaraja, H. N. (2003), Order Statistics, John Wiley & Sons, Hoboken, New Jersey.

Ghitany, M. E., Al-Awadhi, F. A. & Alkhalfan, L. A. (2007), ‘Marshall-Olkin extended Lomax distribution and its application to censored data’, Communications in Statistics-Theory and Methods 36(10), 1855–1866.

Ghitany, M. E., Al-Hussaini, E. K. & Al-Jarallah, R. A. (2005), ‘Marshall-Olkin extended Weibull distribution and its application to censored data’, Journal of Applied Statistics 32(10), 1025–1034.

Kus, C. (2007), ‘A new lifetime distribution’, Computational Statistics & Data Analysis 51(9), 4497–4509.

Lee, E. T. & Wang, J. W. (2003), Statistical Methods for Survival Data Analysis, 3 edn, John Wiley, New York.

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(3), 641–652.

Miller, J. R. (1981), Survival Analysis, John Wiley, New York.

How to Cite

APA

Al-Zahrani, B. and Sagor, H. (2014). The Poisson-Lomax Distribution. Revista Colombiana de Estadística, 37(1), 225–245. https://doi.org/10.15446/rce.v37n1.44369

ACM

[1]
Al-Zahrani, B. and Sagor, H. 2014. The Poisson-Lomax Distribution. Revista Colombiana de Estadística. 37, 1 (Jan. 2014), 225–245. DOI:https://doi.org/10.15446/rce.v37n1.44369.

ACS

(1)
Al-Zahrani, B.; Sagor, H. The Poisson-Lomax Distribution. Rev. colomb. estad. 2014, 37, 225-245.

ABNT

AL-ZAHRANI, B.; SAGOR, H. The Poisson-Lomax Distribution. Revista Colombiana de Estadística, [S. l.], v. 37, n. 1, p. 225–245, 2014. DOI: 10.15446/rce.v37n1.44369. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44369. Acesso em: 28 mar. 2025.

Chicago

Al-Zahrani, Bander, and Hanaa Sagor. 2014. “The Poisson-Lomax Distribution”. Revista Colombiana De Estadística 37 (1):225-45. https://doi.org/10.15446/rce.v37n1.44369.

Harvard

Al-Zahrani, B. and Sagor, H. (2014) “The Poisson-Lomax Distribution”, Revista Colombiana de Estadística, 37(1), pp. 225–245. doi: 10.15446/rce.v37n1.44369.

IEEE

[1]
B. Al-Zahrani and H. Sagor, “The Poisson-Lomax Distribution”, Rev. colomb. estad., vol. 37, no. 1, pp. 225–245, Jan. 2014.

MLA

Al-Zahrani, B., and H. Sagor. “The Poisson-Lomax Distribution”. Revista Colombiana de Estadística, vol. 37, no. 1, Jan. 2014, pp. 225-4, doi:10.15446/rce.v37n1.44369.

Turabian

Al-Zahrani, Bander, and Hanaa Sagor. “The Poisson-Lomax Distribution”. Revista Colombiana de Estadística 37, no. 1 (January 1, 2014): 225–245. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44369.

Vancouver

1.
Al-Zahrani B, Sagor H. The Poisson-Lomax Distribution. Rev. colomb. estad. [Internet]. 2014 Jan. 1 [cited 2025 Mar. 28];37(1):225-4. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44369

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