A new Weibull-G family of distributions

Year 2016, Volume: 45 Issue: 2, 629 - 647, 01.04.2016

M.h. Tahir , M. Zubair M. Mansoor , Gauss M Cordeiro Morad Alizadehk , G. G. Hamedani

Abstract

Statistical analysis of lifetime data is an important topic in reliability
engineering, biomedical and social sciences and others. We introduce
a new generator based on the Weibull random variable called the new
Weibull-G family. We study some of its mathematical properties. Its
density function can be symmetrical, left-skewed, right-skewed, bathtub and reversed-J shaped, and has increasing, decreasing, bathtub,
upside-down bathtub, J, reversed-J and S shaped hazard rates. Some
special models are presented. We obtain explicit expressions for the
ordinary and incomplete moments, quantile and generating functions,
Rényi entropy, order statistics and reliability. Three useful characterizations based on truncated moments are also proposed for the new family. The method of maximum likelihood is used to estimate the model
parameters. We illustrate the importance of the family by means of
two applications to real data sets. 

Keywords

Generating function, hazard function, moment, reliability function, Rényi entropy, Weibull distribution

References

• Alexander, C., Cordeiro, G.M., Ortega, E.M.M. and Sarabia, J.M. Generalized beta generated distributions, Computational Statistics and Data Analysis 56, 1880–1897, 2012.
• Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G.M., Ortega, E.M.M. and Pescim, R.R. A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications, Hacettepa Journal of Mathematics and Statistics 44, 1491–1512, 2015.
• Aljarrah, M.A., Lee, C. and Famoye, F. On generating T-X family of distributions using quantile functions, Journal of Statistical Distributions and Applications 1, Article 2, 2014.
• Alzaatreh, A., Lee, C. and Famoye, F. A new method for generating families of continuous distributions, Metron 71, 63–79, 2013.
• Alzaatreh, A., Lee, C. and Famoye, F. T-normal family of distributions: A new approach to generalize the normal distribution, Journal of Statistical Distributions and Applications 1, Article 16, 2014.
• Alzaghal, A., Famoye, F. and Lee, C. Exponentiated T-X family of distributions with some applications, International Journal of Probability and Statistics 2, 31–49, 2013.
• Amini, M., MirMostafaee, S.M.T.K. and Ahmadi, J. Log-gamma-generated families of distributions, Statistics 48, 913–932, 2014.
• Bourguignon, M., Silva, R.B. and Cordeiro, G.M. The Weibull–G family of probability distributions, Journal of Data Science 12, 53–68, 2014.
• Bjerkedal, T. Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene 72, 130–148, 1960.
• Chen, G. and Balakrishnan, N. A general purpose approximate goodness-of-fit test, Journal of Quality Technology 27, 154–161, 1995.
• Cordeiro, G.M., Alizadeh, M. and Ortega, E.M.M. The exponentiated half-logistic family of distributions: Properties and applications, Journal of Probability and Statistics Article ID 864396, 21 pages, 2014.
• Cordeiro, G.M. and de Castro, M. A new family of generalized distributions, Journal of Statistical Computation and Simulation 81, 883–893, 2011.
• Cordeiro, G.M. and Nadarajah, S. Closed-form expressions for moments of a class of beta generalized distributions, Brazilian Journal of Probability and Statistics 25, 14–33, 2011.
• Cordeiro, G.M., Ortega, E.M.M. and da Cunha, D.C.C. The exponentiated generalized class of distributions, Journal of Data Science 11, 1–27, 2013.
• Cordeiro, G.M., Ortega, E.M.M., Popović, B.V. and Pescim, R.R. The Lomax generator of distributions: Properties, minification process and regression model, Applied Mathematics and Computation 247, 465–486, 2014.
• Cordeiro, G.M., Silva, G.O. and Ortega, E.M.M. The beta extended Weibull distribution, Journal of Probability and Statistical Science (Taiwan) 10, 15–40, 2012.
• Duncan A.J. Quality Control and Industrial Statistics, fourth edition (Irwin- Homewood, 1974).
• Eugene, N., Lee, C. and Famoye, F. Beta-normal distribution and its applications, Communications in Statistics–Theory and Methods 31, 497–512, 2002.
• Glänzel, W. A characterization theorem based on truncated moments and its application to some distribution families, In: Mathematical Statistics and Probability Theory, Volume B, pp. 75–84 (Reidel, Dordrecht, 1987).
• Gupta, R. C., Gupta, P. I. and Gupta, R. D. Modeling failure time data by Lehmann alternatives, Communications in statistics–Theory and Methods 27, 887–904, 1998.
• Mudholkar, G.S., Srivastava, D.K. and Freimer, M. The exponentiated Weibull family: A reanalysis of the bus-motor failure data. Technometrics 37, 436–445, 1995.
• Nadarajah, S., Cordeiro, G.M. and Ortega, E.M.M. The Zografos-Balakrishnan–G family of distributions: Mathematical properties and applications, Communications in Statistics– Theory and Methods 44, 186–215, 2015.
• Nadarajah, S. and Kotz, S. The exponentiated type distributions, Acta Applicandae Mathematica 92, 97–111, 2006.
• Rezaei, S., Nadarajah, S. and Tahghighnia, N. A new three-parameter lifetime distribution, Statistics 47, 835–860, 2013.
• Rényi, A. On measures of entropy and information, In: 4th Berkeley Symposium on Mathematical Statistics and Probability 1, 547–561, 1961.
• Ristić, M.M. and Balakrishnan, N. The gamma-exponentiated exponential distribution, Journal of Statistical Computation and Simulation 82, 1191–1206, 2012.
• Shannon, C.E. A mathematical theory of communication, Bell System Technical Journal 27, 379–432, 1948.
• Tahir, M.H., Cordeiro, G.M., Alzaatreh, A., Mansoor, M. and Zubair, M. The LogisticX family of distributions and its applications, Communications in Statistics–Theory and Methods (to appear) 2016. Doi: 10.1080/03610926.2014.980516.