Generalized transmuted family of distributions: properties and applications

Abstract

We introduce and study general mathematical properties of a new generator of continuous distributions with two extra parameters called the
Generalized Transmuted Family of Distributions. We investigate the shapes and present some special models. The new density function can be expressed as a linear combination of exponentiated densities in terms of the same baseline distribution. We obtain explicit expressions for the ordinary and incomplete moments and generating function, Bonferroni and Lorenz curves, asymptotic distribution of the extreme values, Shannon and Rényi entropies and order statistics, which hold for any baseline model. Further, we introduce a bivariate extension of the new family. We discuss the different methods of estimation of the model parameters and illustrate the potential application of the model via real data. A brief simulation for evaluating Maximum likelihood estimator is done. Finally certain characterziations of our model are presented.

Keywords

• Transmuted distribution,Generated family,Maximum likelihood,Moment,Order statistic,Quantile function,Rényi entropy,Characterizations

References

1. Alexander, C., Cordeiro, G. M., Ortega, E. M., & Sarabia, J. M. (2012). Generalized betagenerated distributions. Computational Statistics & Data Analysis, 56(6), 1880-1897.
2. Alizadeh, M., Cordeiro, G. M., de Brito, E., & Demétrio, C. G. B. (2015). The beta Marshall-Olkin family of distributions. Journal of Statistical Distributions and Applica- tions, 2(1), 1-18.
3. Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G. M., Ortega, E. M., & Pescim, R. R. (2015). A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications. Hacettepe Journal of Mathematics and Statistics (to appear).
4. Alizadeh, M., Tahir, M. H., Cordeiro, G. M., Mansoor, M., Zubair, M., & Hamedani, G. G. (2015). The Kumaraswamy Marshal-Olkin family of distributions. Journal of the Egyptian Mathematical Society.
5. Alzaatreh, A., Lee, C., & Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1), 63-79.
6. Alzaghal, A., Famoye, F., & Lee, C. (2013). Exponentiated T-X Family of Distributions with Some Applications. International Journal of Statistics and Probability, 2(3), p31.
7. Amini, M., MirMostafaee, S. M. T. K., Ahmadi, J. (2014). Log-gamma-generated families of distributions. Statistics, 48(4), 913-932.
8. Andrews, D. F., Herzberg, A. M. (2012). Data: a collection of problems from many elds for the student and research worker. Springer Science Business Media.

9. Aryal, G. R. (2013). Transmuted log-logistic distribution. Journal of Statistics Applications & Probability, 2(1), 11-20.
10. Aryal, G. R., & Tsokos, C. P. (2009). On the transmuted extreme value distribution with application. Nonlinear Analysis: Theory, Methods & Applications, 71(12), e1401-e1407.
11. Aryal, G. R., & Tsokos, C. P. (2011). TransmutedWeibull Distribution: A Generalization of theWeibull Probability Distribution. European Journal of Pure and Applied Mathematics, 4(2), 89-102.
12. Bourguignon, M., Silva, R. B., & Cordeiro, G. M. (2014). The WeibullG family of probability distributions. Journal of Data Science, 12(1), 53-68.
13. Cheng, R. C. H., & Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society. Series B (Methodological), 394-403.
14. Cordeiro, G. M., Alizadeh, M., & Diniz Marinho, P. R. (2015). The type I half-logistic family of distributions. Journal of Statistical Computation and Simulation, (ahead-of-print), 1-22.
15. Cordeiro, G. M., Alizadeh, M., & Ortega, E. M. (2014). The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics, 2014.
16. Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898.
17. Cordeiro, G. M., & Nadarajah, S. (2011). Closed-form expressions for moments of a class of beta generalized distributions.Brazilian journal of probability and statistics, 25(1), 14-33.
18. Cordeiro, G. M., Ortega, E. M., & da Cunha, D. C. (2013). The exponentiated generalized class of distributions. Journal of Data Science, 11(1), 1-27.
19. Cordeiro, G. M., Ortega, E. M., Popovi¢, B. V., & Pescim, R. R. (2014). The Lomax generator of distributions: Properties, minication process and regression model. Applied Mathematics and Computation, 247, 465-486.
20. Eugene, N., Lee, C., & Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31(4), 497-512.
21. Glänzel, W., Teles, A., Schubert, A. (1984). Characterization by truncated moments and its application to Pearson-type distributions. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 66(2), 173-183. Chicago
22. Glänzel, W. (1987). A characterization theorem based on truncated moments and its application to some distribution families. In Mathematical statistics and probability theory (pp. 75-84). Springer Netherlands.
23. Glänzel, W. (1990). Some consequences of a characterization theorem based on truncated moments. Statistics, 21(4), 613-618.
24. Glänzel, W., Hamedani, G. G. (2001). Characterization of univariate continuous distributions. Studia Scientiarum Mathematicarum Hungarica, 37(1-2), 83-118
25. Gupta, R. C., & Gupta, R. D. (2007). Proportional reversed hazard rate model and its applications. Journal of Statistical Planning and Inference, 137(11), 3525-3536.
26. Gupta, R. D., & Kundu, D. (1999). Theory & methods: Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41(2), 173-188.
27. Gupta, R. D., & Kundu, D. (2001). Generalized exponential distribution: dierent method of estimations. Journal of Statistical Computation and Simulation, 69(4), 315-337.
28. Hamedani, G. G. (2002). Characterizations of univariate continuous distributions. II. Studia Scientiarum Mathematicarum Hungarica, 39(3-4), 407-424.
29. Hamedani, G. G. (2006). Characterizations of univariate continuous distributions, III. Stu- dia Scientiarum Mathematicarum Hungarica, 43(3), 361-385.
30. Hamedani, G. (2009). Characterizations of continuous univariate distributions based on the truncated moments of functions of order statistics. Studia Scientiarum Mathematicarum Hungarica, 47(4), 462-484.
31. Jones, M. C. (2004). Families of distributions arising from distributions of order statistics. Test, 13(1), 1-43.
32. Kundu, D., & Gupta, R. D. (2013). Power-normal distribution. Statistics, 47(1), 110-125.
33. 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.
34. Merovci, F. (2013). Transmuted exponentiated exponential distribution. Mathematical Sci- ences And Applications E-Notes, 1(2):112122.
35. Merovci, F. (2014). Transmuted generalized Rayleigh distribution. Journal of Statistics Applications and Probability, 3(1), 9-20.
36. Merovci, F. (2013). Transmuted lindley distribution. International Journal of Open Prob- lems in Computer Science Mathematics, 6.
37. Merovci, F., & Elbatal, I. (2013). Transmuted Lindley-geometric distribution and its applications. arXiv preprint arXiv:1309.3774.
38. Merovci, F. (2013). Transmuted Rayleigh distribution. Austrian Journal of Statistics, 42(1), 21-31.
39. Merovci, F., & Elbatal, I. (2013). Transmuted Lindley-geometric distribution and its applications. arXiv preprint arXiv:1309.3774.
40. Merovci, F., & Elbatal, I. (2014). Transmuted Weibull-geometric distribution and its applications. School of Mathematics Northwest University Xi'an, Shaanxi, PR China, 10(1), 68-82.
41. Merovci, F., & Pukab, L. (2014). Transmuted Pareto Distribution. ProbStat, 7(1):111.
42. Mirhossaini, S. M., & Dolati, A. (2008). On a New Generalization of the Exponential Distribution. Journal of Mathematical Extension.
43. Mudholkar, G. S., & Hutson, A. D. (1996). The exponentiated Weibull family: some properties and a ood data application. Communications in StatisticsTheory and Methods, 25(12), 3059-3083.
44. Mudholkar, G. S., Srivastava, D. K., & Kollia, G. D. (1996). A generalization of theWeibull distribution with application to the analysis of survival data. Journal of the American Statistical Association, 91(436), 1575-1583.
45. Nadarajah, S., & Kotz, S. (2006). The beta exponential distribution.Reliability engineering & system safety, 91(6), 689-697.
46. Nadarajah, S., & Kotz, S. (2006). The exponentiated type distributions. Acta Applicandae Mathematica, 92(2), 97-111.
47. Risti¢, M. M., & Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82(8), 1191-1206.
48. Rényi, A. L. F. R. P. E. D. (1961). On measures of entropy and information. In Fourth Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 547-561).
49. Shannon, C. E. (2001). A mathematical theory of communication. ACM SIGMOBILE Mobile Computing and Communications Review, 5(1), 3-55.
50. 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.
51. Tahir, M. H., Zubair, M., Mansoor, M., Cordeiro, G. M., Alizadeh, M., Hamedani, G. G. (2015). A new Weibull-G family of distributions. Hacet. J. Math. Stat. forthcoming.
52. Tahir, M. H., Cordeiro, G. M., Alzaatreh, A. Y. M. A. N., Mansoor, M., & Zubair, M. (2015). The Logistic-X family of distributions and its applications. Commun. Stat. Theory Methods (2015a). forthcoming.
53. Tahir, M. H., Cordeiro, G. M., Alizadeh, M., Mansoor, M., Zubair, M., Hamedani, G. G. (2015). The odd generalized exponential family of distributions with applications. Journal of Statistical Distributions and Applications, 2(1), 1-28.
54. Team, R. C. (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2012.
55. Torabi, H., & Hedesh, N. M. (2012). The gamma-uniform distribution and its applications. Kybernetika, 48(1), 16-30.
56. Zografos, K., & Balakrishnan, N. (2009). On families of beta-and generalized gammagenerated distributions and associated inference. Statistical Methodology, 6(4), 344-362.
57. Zografos, K., Balakrishnan, N. (2009). On families of beta and generalized gamma generated distributions and associated inference. Statistical Methodology, 6(4), 344-362.