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Revista Colombiana de Estadística

versão impressa ISSN 0120-1751

Rev.Colomb.Estad. vol.42 no.1 Bogotá jan./jun. 2019  Epub 23-Maio-2019 

Artículos originales de investigación

The Exponentiated Kumaraswamy-G Class: General Properties and Application

La clase Kumaraswamy-G exponenciada: propiedades generales y aplicación

Ronaldo Silvaa  , Frank Gomes-Silvab  , Manoel Ramosc  , Gauss Cordeirod  , Pedro Marinhoe  , Thiago A. N. De Andradef 

a Recife Military School, Recife, Brazil.

b Department of Statistics and Informatics, Federal Rural University of Pernambuco, Recife, Brazil.

c Federal Institute of Paraíba, João Pessoa, Brazil.

d Department of Statistics, Federal University of Pernambuco, Recife, Brazil.

e Department of Statistics, Federal University of Paraíba, João Pessoa, Brazil.

f Department of Statistics, Federal University of Pernambuco, Recife, Brazil.


We propose a new family of distributions called the exponentiated Kumaraswamy-G class with three extra positive parameters, which generalizes the Cordeiro and de Castro's family. Some special distributions in the new class are discussed. We derive some mathematical properties of the proposed class including explicit expressions for the quantile function, ordinary and incomplete moments, generating function, mean deviations, reliability, Rényi entropy and Shannon entropy. The method of maximum likelihood is used to fit the distributions in the proposed class. Simulations are performed in order to assess the asymptotic behavior of the maximum likelihood estimates. We illustrate its potentiality with applications to two real data sets which show that the extended Weibull model in the new class provides a better fit than other generalized Weibull distributions.

Key words: BFGS method; Exponential distribution; Exponentiated Kumaraswamy-G; Kumaraswamy distribution; Maximum likelihood estimation


Proponemos una nueva clase de distribuciones llamada la clase de Kumaraswamy-G exponenciada con tres parámetros positivos adicionales, que generaliza la familia de Cordeiro y de Castro. Se discuten algunas distribuciones especiales en la nueva clase. Derivamos algunas propiedades matemáticas de la clase propuesta, incluyendo expresiones explícitas para la función cuartil, momentos ordinarios e incompletos, función generadora, desviaciones medias, confiabilidad, entropía de Rényi y entropía de Shannon. El método de máxima verosimilitud se utiliza para ajustar las distribuciones en la clase propuesta. Se realizaron simulaciones para evaluar el comportamiento asintótico de las estimaciones de máxima verosimilitud. Ilustramos su potencialidad con dos aplicaciones a dos conjuntos de datos reales que muestra que el modelo extendido de Weibull en la nueva clase proporciona un mejor ajuste que otras distribuciones generalizadas de Weibull.

Palabras-clave: Distribución exponencial; Distribución Kumaraswamy; Estimación de máxima verosimilitud; Kumaraswamy-G Exponenciada; Método BFGS

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Alexander, C., Cordeiro, G. M., Ortega, E. M. M. & Sarabia, J. M. (2012), 'Generalized beta-generated distributions', Computational Statistics and Data Analysis 56, 1880-1897. [ Links ]

Alizadeh, M., Tahir, M. H., Cordeiro, G. M., Zubair, M. & Hamedani, G. G. (2015), 'The Kumaraswamy Marshal-Olkin family of distributions', Journal of the Egyptian Mathematical Society 23, 546-557. [ Links ]

Alzaghal, A., Felix, F. & Carl, L. (2013), 'Exponentiated T-X family of distributions with some applications', International Journal of Statistics and Probability 2, 31-49. [ Links ]

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, New York. [ Links ]

Azzalini, A. (1985), 'A class of distributions which includes the normal ones', Scandinavian Journal of Statistics 12, 171-178. [ Links ]

Bagdonavicius, V. & Nikulin, M. (2002), Accelerated life models: modeling and statistical analysis, Chapman and Hall/CRC, Boca Raton. [ Links ]

Barlow, R., Toland, R. & Freeman, T. (1984), A bayesian analysis of stress-rupture life of kevlar 49/epoxy spherical pressure vessels, Conference on Applications of Statistics, Marcel Dekker, New York. [ Links ]

Bebbington, M., Lai, C. D. & Zitikis, R. (2007), 'A flexible Weibull extension', Reliability Engineering and System Safety 92, 719-726. [ Links ]

Chen, G. & Balakrishnan, N. (1995), 'A general purpose approximate goodnessof-fit test', Journal of Quality Technology 27, 154-161. [ 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, 1323-1337. [ Links ]

Cordeiro, G. M., Alizadeh, M., Tahir, M. H., Mansoor, M., Bourguignon, M. & G., H. G. (2017), 'The generalized odd log logistic family of distributions: properties, regression models and applications', Journal of Statistical Computation and Simulation 87, 908-932. [ Links ]

Cordeiro, G. M. & de Castro, M. (2011), 'A new family of generalized distributions', Journal of Statistical Computation and Simulation 81, 883-893. [ Links ]

Cordeiro, G. M., Ortega, E. M. M. & Cunha, D. C. C. (2013), 'The exponentiated generalized class of distributions', Journal of Data Science 11, 1-27. [ Links ]

Cordeiro, G. M., Silva, G. O. & Ortega, E. M. M. (2012), 'The beta extended Weibull distribution', Journal of Probability and Statistical Science 10, 15-40. [ Links ]

Cox, D. R. & Hinkley, D. V. (1974), Theoretical Statistics, Chapman and Hall, London. [ Links ]

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

Gomes-Silva, F., Percontini, A., Brito, E., Ramos, M. W., Silva, R. V. & Cordeiro, G. M. (2017), 'The odd Lindley-G family of distributions', Austrian Journal of Statistics 46, 65-87. [ Links ]

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

Gupta, R. C., Gupta, R. D. & Gupta, P. L. (1998), 'Modeling failure time data by Lehman alternatives', Communications in Statistics Theory and Methods 27, 887-904. [ Links ]

Gupta, R. D. & Kundu, D. (1999), 'Generalized exponential distributions', Australian and New Zealand Journal of Statistics 41, 173-188. [ 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, 1-18. [ Links ]

Jones, M. C. (2004), 'Families of distributions arising from the distributions of order statistics', Test 13, 1-43. [ Links ]

Kenney, J. F. & Keeping, E. S. (1962), Mathematics of Statistics, Princeton, Nueva York. [ Links ]

Lehmann, E. L. (1953), 'The power of rank tests', The Annals of Mathematical Statistics 24, 23-43. [ Links ]

Lemonte, A. J. (2013), 'A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function', Computational Statistics and Data Analysis 62, 149-170. [ Links ]

Lemonte, A. J., Barreto-Souza, W. & Cordeiro, G. M. (2013), 'The exponentiated Kumaraswamy distribution and its log-transform', Brazilian Journal of Probability and Statistics 27, 31-53. [ Links ]

Marshall, A. N. & Olkin, I. (1997), 'A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families', Biometrika 84, 641-552. [ Links ]

Meshkat, R. S., Torabi, H. & Hamedani, G. G. (2016), 'A generalized gamma-Weibull distribution: model, properties and applications', Pakistan Journal of Statistics and Operation Research 12, 201-212. [ Links ]

Moors, J. J. A. (1988), 'A quantile alternative for kurtosis', Journal of the Royal Statistical Society D 37, 25-32. [ Links ]

Mudholkar, G. S. & Srivastava, D. K. (1993), 'Exponentiated Weibull family for analyzing bathtub failure-rate data', IEEE Transactions on Reliability 42, 299-302. [ Links ]

Mudholkar, G. S., Srivastava, D. K. & Kollia, G. D. (1996), 'A generalization of the Weibull distribution with application to the analysis of survival data', Journal of American Statistical Association 91, 1575-1583. [ Links ]

Nadarajah, S. (2005), 'The exponentiated Gumbel distribution with climate application', Environmetrics 17, 13-23. [ Links ]

Nadarajah, S., Cancho, V. G. & Ortega, E. M. M. (2013), 'The geometric exponential Poisson distribution', Statistical Methods & Applications 22, 355-380. [ Links ]

Nadarajah, S. & Gupta, A. K. (2007), 'The exponentiated gamma distribution with application to drought data', Calcutta Statistical Association Bulletin 59, 29-54. [ Links ]

Nadarajah, S. & Haghighi, F. (2011), 'An extension of the exponential distribution', Statistics 45, 543-558. [ Links ]

Nadarajah, S., Jayakumar, K. & Ristic, M. M. (2013), 'A new family of lifetime models', Journal of Statistical Computation and Simulation 83, 1389-1404. [ Links ]

Nadarajah, S. & Kotz, S. (2006), 'The exponentiated-type distributions', Acta Applicandae Mathematicae 92, 97-111. [ Links ]

Nofal, Z. M., A-fy, A. Z., Yousof, H. M. & Cordeiro, G. M. (2017), 'The generalized transmuted-G family of distributions', Commun Stat Theory Methods 46, 4119-4136. [ Links ]

Pescim, R. R., Cordeiro, G. M., Demetrio, C. G. B., Ortega, E. M. M. & Nadarajah, S. (2012), 'The new class of Kummer beta generalized distributions', SORT 36, 153-180. [ Links ]

R Development Core Team, R: A Language and Environment for Statistical Computing (2012), R Foundation for Statistical Computing, Austria. [ Links ]

Rényi, A. (1961), On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley. [ Links ]

Ristic, M. M. & Balakrishnan, N. (2012), 'The gamma-exponentiated exponential distribution', Journal of Statistical Computation and Simulation 82, 1191-1206. [ Links ]

Rodrigues, J. A. & Silva, A. P. (2015), 'The exponentiated Kumaraswamyexponential distribution', British Journal of Applied Science & Technology 10, 1-12. [ Links ]

Rodrigues, J. A., Silva, A. P. & Hamedani, G. G. (2016), 'The exponentiated Kumaraswamy inverse Weibull distribution with application in survival analysis', Journal of Statistical Theory and Applications 15, 8-24. [ Links ]

Shannon, C. E. (1951), 'Prediction and entropy of printed english', The Bell System Technical Journal 30, 50-64. [ Links ]

Silva, R. V., Andrade, T. A. N., Maciel, D. B. M., Campos, R. P. S. & Cordeiro, G. M. (2013), 'A new lifetime model: The gamma extended Fréchet distribution', Journal of Statistical Theory and Applications 12, 39-54. [ Links ]

Tahir, M. H. & Nadarajah, S. (2015), 'Parameter induction in continuous univariate distributions: Well-established G families', Annals of the Brazilian Academy of Sciences 87, 539-568. [ Links ]

Torabi, H. & Montazari, N. H. (2014), 'The logistic-uniform distribution and its application', Communications in Statistics - Simulation and Computation 43, 2551-2569. [ Links ]

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

Zografos, K. & Balakrishnan, N. (2009), 'On families of beta-and generalized gamma-generated distribution and associate inference', Statistical Methodology 6, 344-362. [ Links ]

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