Published

2017-01-16

Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution

Distribución Singh-Maddala trams,utada: una nueva distribución, flexible y con forma de bañera invertida para la función de riesgo

DOI:

https://doi.org/10.15446/rce.v40n1.50085

Keywords:

Moments, Parameter Estimation, Transmuted Singh-Maddala Distribution, TL-Moments, Upsidedown Bathtub Shaped Hazard Rate (en)
distribución Singh-Maddala transmuetada, función de riesgo invertida, momentos, momentos TL, estimación de parámetros. (es)

Downloads

Authors

  • Mirza Naveed Shahzad University of Gujrat
  • Faton Merovci Department of Mathematics, University of Prishtina "Hasan Prishtina", Republic of Kosovo
  • Zahid Asghar Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan

The Singh-Maddala distribution is very popular to analyze the data on income, expenditure, actuarial, environmental, and reliability related studies. To enhance its scope and application, we propose four parameters transmuted
Singh-Maddala distribution, in this study. The proposed distribution is relatively more flexible than the parent distribution to model a variety of data sets. Its basic statistical properties, reliability function, and behaviors of the hazard function are derived. The hazard function showed the decreasing and an upside-down bathtub shape that is required in various survival analysis. The order statistics and generalized TL-moments with their special cases such as L-, TL-, LL-, and LH-moments are also explored. Furthermore, the maximum likelihood estimation is used to estimate the unknown parameters of the transmuted Singh-Maddala distribution. The real data sets are considered to illustrate the utility and potential of the proposed model. The results indicate that the transmuted Singh-Maddala distribution models the datasets better than its parent distribution.

La distribución Singh-Maddala es muy popular para analizar datos de
ingresos, gastos, actuariales, ambientales y de confiabilidad.
Para mejorar su alcance y aplicación se propone su extensión a la distribución de cuatro parámetros Singh-Maddala transmutada. Esta es más flexible en la modelación de diversos conjuntos de datos. Sus propiedades básicas, las funciones de confiabilidad y riesgos son estudiadas. La función de riesgo es decrecientes o tiene forma de bañera invertida. Como se requiere en varios estudios de sobrevivencia se exploran sus estadísticas de orden y los momentos TL, con sus casos especiales L, TL, LL y LH. Se emplea máxima verosimilitud para la estimación de los cuatro parámetros. Datos reales son usados para ilustrar la utilidad y potencialidad del modelo propuesto. Los resultados indican que la distribución propuesta ajusta mejor que la original.

https://doi.org/10.15446/rce.v40n1.50085

Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution

Distribución Singh-Maddala transmutada: una nueva distribución, flexible y con forma de bañera invertida para la función de riesgo

MIRZA NAVEED SHAHZAD1, FATON MEROVCI2, ZAHID ASGHAR3

1University of Gujrat, Department of Statistics, Gujrat, Pakistan. PhD. Email: nvd.shzd@uog.edu.pk
2University of Prishtina ''Hasan Prishtina'', Pristina, Republic of Kosovo. PhD. Email: fmerovci@yahoo.com
3Quaid-i-Azam University, Department of Statistics, Islamabad, Pakistan. PhD. Email: g.zahid@gmail.com

Abstract

The Singh-Maddala distribution is very popular to analyze the data on income, expenditure, actuarial, environmental, and reliability related studies. To enhance its scope and application, we propose four parameters transmuted Singh-Maddala distribution, in this study. The proposed distribution is relatively more flexible than the parent distribution to model a variety of data sets. Its basic statistical properties, reliability function, and behaviors of the hazard function are derived. The hazard function showed the decreasing and an upside-down bathtub shape that is required in various survival analysis. The order statistics and generalized TL-moments with their special cases such as L-, TL-, LL-, and LH-moments are also explored. Furthermore, the maximum likelihood estimation is used to estimate the unknown parameters of the transmuted Singh-Maddala distribution. The real data sets are considered to illustrate the utility and potential of the proposed model. The results indicate that the transmuted Singh-Maddala distribution models the datasets better than its parent distribution.

Key words: Moments, Parameter Estimation, Transmuted Singh-Maddala Distribution, TL-Moments, Upsidedown Bathtub Shaped Hazard Rate.

Resumen

La distribución Singh-Maddala es muy popular para analizar datos de ingresos, gastos, actuariales, ambientales y de confiabilidad. Para mejorar su alcance y aplicación se propone su extensión a la distribución de cuatro parámetros Singh-Maddala transmutada. Esta es más flexible en la modelación de diversos conjuntos de datos. Sus propiedades básicas, las funciones de confiabilidad y riesgos son estudiadas. La función de riesgo es decrecientes o tiene forma de bañera invertida. Como se requiere en varios estudios de sobrevivencia se exploran sus estadísticas de orden y los momentos TL, con sus casos especiales L, TL, LL y LH. Se emplea máxima verosimilitud para la estimación de los cuatro parámetros. Datos reales son usados para ilustrar la utilidad y potencialidad del modelo propuesto. Los resultados indican que la distribución propuesta ajusta mejor que la original.

Palabras clave: distribución Singh-Maddala transmuetada, función de riesgo invertida, momentos, momentos TL, estimación de parámetros.

Texto completo disponible en PDF

References

1. Ahmad, A., Ahmad, S. P. & Ahmed, A. (2014), 'Transmuted Inverse Rayleigh Distribution: A Generalization of the Inverse Rayleigh Distribution', Mathematical Theory and Modeling 4(6), 177-185.

2. Arnold, B. C., Balakrishnan, N. & Nagaraja, H. N. (1992), A First Course in Order Statistics, Vol. 54, Siam.

3. Aryal, G. R. (2013), 'Transmuted log-logistic distribution', Journal of Statistics Applications and Probability 2(1), 11-20.

4. Aryal, G. R. & Tsokos, C. P. (2011), 'Transmuted Weibull distribution: A generalization of the Weibull probability distribution', European Journal of Pure and Applied Mathematics 4(2), 89-102.

5. Balakrishnan, N. & Cohen, A. C. (1991), Order statistics & inference: estimation methods, Academic Press.

6. Bayazit, M. & Onoz, B. (2002), 'LL-moments for estimating low fow quantiles', Hydrological Sciences Journal 47(5), 707-720.

7. Brzezinski, M. (2014), 'Empirical modeling of the impact factor distribution', Journal of Informetrics 8(2), 362-368.

8. Elamir, E. A. & Seheult, A. H. (2003), 'Trimmed L-moments', Computational Statistics and Data Analysis 43(3), 299-314.

9. Elbatal, I. (2013), 'Transmuted generalized inverted exponential distribution', Economic Quality Control 28(2), 125-133.

10. Hosking, J. R. M. (1990), 'L-moments: analysis and estimation of distributions using linear combinations of order statistics', Journal of the Royal Statistical Society. Series B 52, 105-124.

11. Khan, M. S. & King, R. (2014), 'A new class of transmuted inverse Weibull Distribution for reliability analysis', American Journal of Mathematical and Management Sciences 33(4), 261-286.

12. Kleiber, C. & Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Vol. 470, John Wiley & Sons.

13. Lee, E. & Wang, J. (2003), Statistical Methods for Survival Data Analysis, Wiley, New York.

14. Merovci, F. (2013), 'Transmuted Rayleigh distribution', Austrian Journal of Statistics 42(1), 21-31.

15. Sakulski, D., Jordaan, A., Tin, L. & Greyling, C. (2014), Fitting theoretical distributions to Rainy Days for Eastern Cape Drought Risk assessment, 'Proceedings of DailyMeteo. org/2014 Conference', p. 48.

16. Shahzad, M. N. & Asghar, Z. (2016), 'Transmuted Dagum Distribution: A more flexible and broad shaped hazard function model', Hacettepe Journal of Mathematics and Statistics 45(1), 1-18.

17. Shao, Q., Wang, Q. & Zhang, L. (2013), A stochastic weather generation method for temporal precipitation simulation, '20th international congress on modelling and simulation', Society of Australia and New Zealand, , , p. 2681-2687.

18. Sharma, V. K., Singh, S. K. & Singh, U. (2014), 'A new upside-down bathtub shaped hazard rate model for survival data analysis', Applied Mathematics and Computation 239, 242-253.

19. Shaw, W. T. & Buckley, I. R. (2009), 'A stochastic weather generation method for temporal precipitation simulation: The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map', arXiv preprint arXiv:0901.0434, 1-8.

20. Singh, S. K. & Maddala, G. (1976), 'A function for the size distribution and incomes', Econometrica 44, 963-970.

21. Wang, Q. J. (1997), 'LH moments for statistical analysis of extreme events', Water Resources Research 33(12), 2841-2848.

22. Zimmer, W. J., Keats, J. B. & Wang, F. K. (1998), 'The Burr XII distribution in reliability analysis', Journal of Quality Technology 30(4), 386-394.

[Recibido en abril de 2015. Aceptado en febrero de 2016]

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

@ARTICLE{RCEv40n1a01,
    AUTHOR  = {Naveed Shahzad, Mirza and Merovci, Faton and Asghar, Zahid},
    TITLE   = {{Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2017},
    volume  = {40},
    number  = {1},
    pages   = {1-27}
}

References

Ahmad, A., Ahmad, S. P. & Ahmed, A. (2014), ‘Transmuted Inverse Rayleigh Distribution: A Generalization of the Inverse Rayleigh Distribution’, Mathematical Theory and Modeling 4(6), 177–185.

Arnold, B. C., Balakrishnan, N. & Nagaraja, H. N. (1992), A First Course in Order Statistics, Vol. 54, Siam.

Aryal, G. R. (2013), ‘Transmuted log-logistic distribution’, Journal of Statistics Applications and Probability 2(1), 11–20.

Aryal, G. R. & Tsokos, C. P. (2011), ‘Transmuted Weibull distribution: A generalization of the Weibull probability distribution’, European Journal of Pure and Applied Mathematics 4(2), 89–102.

Balakrishnan, N. & Cohen, A. C. (1991), Order statistics & inference: estimation methods, Academic Press.

Bayazit, M. & Onoz, B. (2002), ‘LL-moments for estimating low fow quantiles’, Hydrological Sciences Journal 47(5), 707–720.

Brzezinski, M. (2014), ‘Empirical modeling of the impact factor distribution’, Journal of Informetrics 8(2), 362–368.

Elamir, E. A. & Seheult, A. H. (2003), ‘Trimmed L-moments’, Computational Statistics and Data Analysis 43(3), 299–314.

Elbatal, I. (2013), ‘Transmuted generalized inverted exponential distribution’, Economic Quality Control 28(2), 125–133.

Hosking, J. R. M. (1990), ‘L-moments: analysis and estimation of distributions using linear combinations of order statistics’, Journal of the Royal Statistical Society. Series B 52, 105–124.

Khan, M. S. & King, R. (2014), ‘A new class of transmuted inverse Weibull Distribution for reliability analysis’, American Journal of Mathematical and Management Sciences 33(4), 261–286.

Khan, M. S., King, R. & Hudson, I. (2014), ‘Characterizations of the transmuted inverse Weibull distribution’, ANZIAM Journal 55, 197–217.

Kleiber, C. & Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Vol. 470, John Wiley & Sons.

Lee, E. & Wang, J. (2003), Statistical Methods for Survival Data Analysis, Wiley, New York.

Merovci, F. (2013), ‘Transmuted Rayleigh distribution’, Austrian Journal of Statistics 42(1), 21–31.

Sakulski, D., Jordaan, A., Tin, L. & Greyling, C. (2014), Fitting theoretical distributions to Rainy Days for Eastern Cape Drought Risk assessment, in ‘Proceedings of DailyMeteo. org/2014 Conference’, p. 48.

Shahzad, M. N. & Asghar, Z. (2016), ‘Transmuted Dagum Distribution: A more flexible and broad shaped hazard function model’, Hacettepe Journal of Mathematics and Statistics 45(1), 1–18.

Shao, Q., Wang, Q. & Zhang, L. (2013), A stochastic weather generation method for temporal precipitation simulation, in ‘20th international congress on modelling and simulation’, Society of Australia and New Zealand, pp. 2681–2687.

Sharma, V. K., Singh, S. K. & Singh, U. (2014), ‘A new upside-down bathtub shaped hazard rate model for survival data analysis’, Applied Mathematics and Computation 239, 242–253.

Shaw, W. T. & Buckley, I. R. (2009), ‘A stochastic weather generation method for temporal precipitation simulation: The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map’, arXiv preprint arXiv:0901.0434 pp. 1–8.

Singh, S. K. & Maddala, G. (1976), ‘A function for the size distribution and incomes’, Econometrica 44, 963–970.

Wang, Q. J. (1997), ‘LH moments for statistical analysis of extreme events’, Water Resources Research 33(12), 2841–2848.

Zimmer, W. J., Keats, J. B. & Wang, F. K. (1998), ‘The Burr XII distribution in reliability analysis’, Journal of Quality Technology 30(4), 386–394.

How to Cite

APA

Shahzad, M. N., Merovci, F. and Asghar, Z. (2017). Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution. Revista Colombiana de Estadística, 40(1), 1–27. https://doi.org/10.15446/rce.v40n1.50085

ACM

[1]
Shahzad, M.N., Merovci, F. and Asghar, Z. 2017. Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution. Revista Colombiana de Estadística. 40, 1 (Jan. 2017), 1–27. DOI:https://doi.org/10.15446/rce.v40n1.50085.

ACS

(1)
Shahzad, M. N.; Merovci, F.; Asghar, Z. Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution. Rev. colomb. estad. 2017, 40, 1-27.

ABNT

SHAHZAD, M. N.; MEROVCI, F.; ASGHAR, Z. Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution. Revista Colombiana de Estadística, [S. l.], v. 40, n. 1, p. 1–27, 2017. DOI: 10.15446/rce.v40n1.50085. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/50085. Acesso em: 25 apr. 2024.

Chicago

Shahzad, Mirza Naveed, Faton Merovci, and Zahid Asghar. 2017. “Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution”. Revista Colombiana De Estadística 40 (1):1-27. https://doi.org/10.15446/rce.v40n1.50085.

Harvard

Shahzad, M. N., Merovci, F. and Asghar, Z. (2017) “Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution”, Revista Colombiana de Estadística, 40(1), pp. 1–27. doi: 10.15446/rce.v40n1.50085.

IEEE

[1]
M. N. Shahzad, F. Merovci, and Z. Asghar, “Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution”, Rev. colomb. estad., vol. 40, no. 1, pp. 1–27, Jan. 2017.

MLA

Shahzad, M. N., F. Merovci, and Z. Asghar. “Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution”. Revista Colombiana de Estadística, vol. 40, no. 1, Jan. 2017, pp. 1-27, doi:10.15446/rce.v40n1.50085.

Turabian

Shahzad, Mirza Naveed, Faton Merovci, and Zahid Asghar. “Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution”. Revista Colombiana de Estadística 40, no. 1 (January 1, 2017): 1–27. Accessed April 25, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/50085.

Vancouver

1.
Shahzad MN, Merovci F, Asghar Z. Transmuted Singh-Maddala Distribution: A new Flexible and Upside-Down Bathtub Shaped Hazard Function Distribution. Rev. colomb. estad. [Internet]. 2017 Jan. 1 [cited 2024 Apr. 25];40(1):1-27. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/50085

Download Citation

CrossRef Cited-by

CrossRef citations1

1. Mariam Zahid. (2018). Stability within family of Pareto models. Biometrics & Biostatistics International Journal, 7(3) https://doi.org/10.15406/bbij.2018.07.00207.

Dimensions

PlumX

Article abstract page views

674

Downloads

Download data is not yet available.

License

Copyright (c) 2017 Revista Colombiana de Estadística

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).