## Indicators

• Similars in SciELO

## Print version ISSN 0120-1751

### Rev.Colomb.Estad. vol.41 no.1 Bogotá Jan./June 2018

#### http://dx.doi.org/10.15446/rce.v41n1.66542

Artículos originales de investigación

The Odd Log-Logistic Dagum Distribution: Properties and Applications

La distribución Odd Log-Logistica de Dagum: propiedades y aplicaciones

Filippo Domma1  a  , Abbas Eftekharian2  b  , Ahmed Z. Afify3  c  , Morad Alizadeh4  d  , Indranil Ghosh5  e

1 Department of Economics, Statistics and Finance, University of Calabria, Rende CS, Italy.

2 Department of Statistics, University of Hormozgan, Bandar Abbas, Iran.

3 Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt.

4 Department of Statistics, Faculty of Sciences, Persian Gulf University, Bushehr, Iran.

5 Department of Mathematics and Statistics, University of North Carolina Wilmington city, United States.

Abstract

This paper introduces a new four-parameter lifetime model called the odd log-logistic Dagum distribution. The new model has the advantage of being capable of modeling various shapes of aging and failure criteria. We derive some structural properties of the model odd log-logistic Dagum such as order statistics and incomplete moments. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimation are discussed. We prove empirically the importance and flexibility of the new model in modeling real data.

Key words: Dagum Distribution; Maximum Likelihood; Odd Log-Logistic Family; Order Statistics

Resumen

Este artículo introduce un nuevo modelo de sobrevida de cuatro parámetros llamado la distribución Odd Log-Logistica de Dagum. El nuevo modelo tiene la ventaja de ser capaz de modelar varias formas de envejecimiento y criterios de falla. Derivamos propiedades estructurales del modelo Odd Log-Logistica de Dagum tales como estadísticos de orden y momentos incompletos. El método de máxima verosimilitud es usado para estimar los parámetros del modelo. Se discuten resultados de algunas simulaciones, las cuales permitieron establecer la eficiencia del método de máxima verosimilitud. Probamos empíricamente la importancia y flexibilidad del nuevo modelo a través de un ejemplo en datos reales.

Palabras-clave: distribución Dagum; familia Odd-Log-Logística; máxima verosimilitud; estadísticas de orden

Full text available only in PDF format.

References

Andrews, D. F. & Herzberg, A. M. (2012), Data: a collection of problems from many fields for the student and research worker, Springer Science & Business Media. [ Links ]

Armitage, P. & Berry, G. (1987), Distribution-free methods, in 'Statistical methods in medical research', Blackwell Scientific Oxford, London, Edinburgh, Boston, Melbourne, pp. 408-420. [ Links ]

Barlow, R., Toland, R. & Freeman, T. (1984), A bayesian analysis of stress-rupture life of kevlar 49/epoxy spherical pressure vessels, in 'Proc. Conference on Applications of Statistics, Marcel Dekker, New York'. [ Links ]

Cooray, K. & Ananda, M. M. (2008), 'A generalization of the half-normal distribution with applications to lifetime data', Communications in Statistics-Theory and Methods 37(9), 1323-1337. [ Links ]

Da Cruz, J., Cordeiro, G., Ortega, E. & Alizadeh, M. (2014), 'A new family of distributions: Theory and applications in survival analysis', Statistics . [ Links ]

Dagum, C. (1977), 'New model of personal income-distribution-specification and estimation', Economie appliquée 30(3), 413-437. [ Links ]

Domma, F. (2002), 'L andamento della hazard function nel modello di Dagum a tre parametri', Quaderni di Statistica 4, 1-12. [ Links ]

Domma, F. (2007), 'Asymptotic distribution of the maximum likelihood estimators of the parameters of the right-truncated Dagum distribution', Communications in Statistics-Simulation and Computation 36(6), 1187-1199. [ Links ]

Domma, F. & Condino, F. (2013), 'The beta-Dagum distribution: definition and properties', Communications in Statistics-Theory and Methods 42(22), 4070-4090. [ Links ]

Domma, F. & Condino, F. (2016), 'Use of the beta-Dagum and beta-singh-maddala distributions for modeling hydrologic data', Stochastic Environmental Research and Risk Assessment pp. 1-15. [ Links ]

Domma, F., Giordano, S. & Zenga, M. (2011), 'Maximum likelihood estimation in Dagum distribution with censored samples', Journal of Applied Statistics 38(12), 2971-2985. [ Links ]

Domma, F., Latorre, G. & Zenga, M. (2012), 'The Dagum distribution in reliability analisys.', Statistica & Applicazioni 10(2). [ Links ]

Elbatal, I. & Aryal, G. (2015), 'Transmuted Dagum distribution with applications', Chilean Journal of Statistics (ChJS) 6(2). [ Links ]

Evans, D. L., Drew, J. H. & Leemis, L. M. (2008), 'The distribution of the Kolmogorov-Smirnov, Cramer-Von Mises, and Anderson-Darling test statistics for exponential populations with estimated parameters', Communications in Statistics-Simulation and Computation 37(7), 1396-1421. [ Links ]

Gleaton, J. & Lynch, J. (2006), 'Properties of generalized log-logistic families of lifetime distributions', Journal of Probability and Statistical Science 4(1), 51-64. [ Links ]

Gradshteyn, I. & Ryzhik, I. (2007), Table of Integrals, Series, and Products, Elsevier/Academic Press. [ Links ]

Huang, S. & Oluyede, B. O. (2014), 'Exponentiated kumaraswamy-Dagum distribution with applications to income and lifetime data', Journal of Statistical Distributions and Applications 1(1), 1-20. [ Links ]

Kenney, J. & Keeping, E. (1962), Mathematics of Statistics, Vol. 1, Princeton, NJ: Van Nostrand. [ Links ]

Kleiber, C. (1996), 'Dagum vs. singh-maddala income distributions', Economics Letters 53(3), 265-268. [ Links ]

Kleiber, C. (1999), 'On the Lorenz order within parametric families of income distributions', Sankhyā: The Indian Journal of Statistics, Series B pp. 514-517. [ Links ]

Kleiber, C. (2008), A guide to the Dagum distributions, in 'Modeling Income Distributions and Lorenz Curves', Springer, pp. 97-117. [ Links ]

Kleiber, C. & Kotz, S. (2003), Statistical size distributions in economics and actuarial sciences, Vol. 470, John Wiley & Sons. [ Links ]

Moors, J. (1988), 'A quantile alternative for kurtosis', The Statistician pp. 25-32. [ Links ]

Navarro, J., Franco, M. & Ruiz, J. (1998), 'Characterization through moments of the residual life and conditional spacings', Sankhyā: The Indian Journal of Statistics, Series A pp. 36-48. [ Links ]

Oluyede, B. O., Motsewabagale, G., Huang, S., Warahena-Liyanage, G. & Pararai, M. (2016), 'The Dagum-Poisson distribution: model, properties and application', Electronic Journal of Applied Statistical Analysis 9(1), 169-197. [ Links ]

Oluyede, B. O. & Rajasooriya, S. (2013), 'The MC-Dagum distribution and its statistical properties with applications', Asian Journal of Mathematics and Applications 2013. [ Links ]

Oluyede, B. O. & Ye, Y. (2014), 'Weighted Dagum and related distributions', Afrika Matematika 25(4), 1125-1141. [ Links ]

Paranaíba, P. F., Ortega, E. M., Cordeiro, G. M. & Pescim, R. R. (2011), 'The beta burr xii distribution with application to lifetime data', Computational Statistics & Data Analysis 55(2), 1118-1136. [ Links ]

Pérez, C. G. & Alaiz, M. P. (2011), 'Using the Dagum model to explain changes in personal income distribution', Applied Economics 43(28), 4377-4386. [ Links ]

Quintano, C. & D'Agostino, A. (2006), 'Studying inequality in income distribution of single-person households in four developed countries', Review of Income and Wealth 52(4), 525-546. [ Links ]

Silverman, B. W. (1986), Density estimation for statistics and data analysis, Vol. 26, CRC press. [ Links ]

Tahir, M., Cordeiro, G. M., Mansoor, M., Zubair, M. & Alizadeh, M. (2016), 'The Weibull-Dagum distribution: properties and applications', Communications in Statistics-Theory and Methods (just-accepted). [ Links ]

Thode, H. C. (2002), Testing for normality, Vol. 164, CRC press. [ Links ]

Torabi, H., Montazeri, N. H. & Grané, A. (2016), 'A test for normality based on the empirical distribution function', SORT 40(1), 3-36. [ Links ]