The complementary geometric transmuted-$G$ family of distributions: model, properties and application

Year 2018, Volume: 47 Issue: 5, 1348 - 1374, 16.10.2018

Ahmed Z. Afify Gauss M. Cordeiro , Saralees Nadarajah Haitham M. Yousof , Gamze Ozel , Zohdy M. Nofal , Emrah Altun

Abstract

We introduce a new family of continuous distributions called the complementary geometric transmuted-$G$ family, which extends the transmuted

family proposed by Shaw and Buckley (2007). Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, quantile and generating functions, entropies, order statistics and probability weighted moments are derived. Two special models of the introduced family are discussed in detail. The maximum likelihood method is used for estimating the model parameters. The importance and flexibility of the new family are illustrated by means of two applications to real data sets. We provide some simulation results to assess the performance of the proposed model.

Keywords

Complementary geometric-G Family, Maximum likelihood, Order statistic, Probability weighted moment, Transmuted family

References

• Afify A. Z., Alizadeh, M., Yousof, H. M., Aryal, G. and Ahmad, M. The transmuted geometric-G family of distributions: Theory and applications. Pakistan Journal of Statistics, 32, 139-160,(2016a).
• Afify A. Z., Cordeiro G. M., Yousof, H. M., Alzaatreh, A. and Nofal, Z. M. The Kumaraswamy transmuted-G family of distributions: Properties and applications. Journal of Data Science, 14, 245-270, (2016b).
• Afify A. Z., Cordeiro G. M., Yousof, H. M., Saboor, A. and Ortega, E. M. M. The Marshall- Olkin additive Weibull distribution with variable shapes for the hazard rate. Hacettepe Journal of Mathematics and Statistics, forthcoming, (2016c).
• Afify A. Z., Yousof, H. M. and Nadarajah, S. The beta transmuted-H family for lifetime data. Statistics and Its Interface, forthcoming, (2016d).
• Al-Babtain, A., Fattah, A. A., Ahmed, A. N. and Merovci, F. The Kumaraswamy transmuted exponentiated modied Weibull distribution. Communications in Statistics- Simulation and Computation, forthcoming, 2015
• Al-Hussaini, E. K. and Ahsanullah, M. Exponentiated Distributions. Volume 5, Springer, 2015.
• 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.
• Alzaatreh, A., Lee, C. and Famoye, F. A new method for generating families of continuous distributions. Metron, 71, 63-79, 2013.
• Ashour, S. K. and Eltehiwy M. A. Transmuted exponentiated modified Weibull distribution. International Journal of Basic and Applied Sciences, 2, 258-269, 2013.
• Bourguignon, M., Silva, R. B. and Cordeiro, G. M. The Weibull-G family of probability distributions. Journal of Data Science, 12, 53-68, 2014.
• Cakmakyapan, S. and Kadilar, G. O. A new customer lifetime duration distribution: The Kumaraswamy Lindley distribution. International Journal of Trade, Economics and Finance, 5, 441-444, 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., Ortega, E. M. M. and Lemonte, A. J. (2014). The exponential-Weibull lifetime distribution. Journal of Statistical Computation and Simulation, 84, 2592-2606, 2014.
• Cordeiro, G. M., Ortega, E. M. M. and Nadarajah, S. The Kumaraswamy Weibull distri- bution with application to failure data. Journal of the Franklin Institute, 347, 1399-1429, 2010.
• Elbatal, I. and Aryal, G. On the transmuted additive Weibull distribution. Austrian Journal of Statistics, 42, 117-132, 2013.
• Eugene, N., Lee, C. and Famoye, F. Beta-normal distribution and its applications. Commu- nications in Statistics-Theory and Methods, 31, 497-512, 2002.
• Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
• Korkmaz, M. C. and Genc, A. I. A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution. Communications in Statistics Simulation and Computation, forthcoming, 2016.
• Lee, C., Famoye, F. and Olumolade, O. Beta-Weibull distribution: Some properties and applications to censored data. Journal of Modern Applied Statistical Methods, 6, 173-186, 2007.
• Louzada-Neto, F., Roman, M. and Cancho, V. G. The complementary exponential-geometric distribution for lifetime data. Computational Statistics and Data Analysis, 55, 2516-2524, 2011.
• Merovci, F., Alizadeh, M., Yousof, H. M. and Hamedani G. G. The exponentiated transmuted-G family of distributions: Theory and applications. Communications in Statistics-Theory and Method, forthcoming, 2016.
• Merovci, F. and Sharma, V. K. The beta Lindley distribution: Properties and applications. Journal of Applied Mathematics, ID 198951, 1-10, 2014.
• Nofal, Z. M., Afify, A. Z., Yousof, H. M. and Cordeiro, G. M. The generalized transmuted-G family of distributions. Communications in Statistics-Theory and Methods, 46, 4119-4136, 2017.
• Ozel, G., Alizadeh, M., Cakmakyapan, S., Hamedani, G. G., Ortega, E. M. M. and Cancho, V. G. The odd log-logistic Lindley Poisson model for lifetime data. Communications in Statistics Theory and Methods, forthcoming, 2016.
• Saboor, A., Elbatal, I. and Cordeiro, G. M. The transmuted exponentiated Weibull geomet- ric distribution: Theory and applications. Hacettepe Journal of Mathematics and Statistics, forthcoming, 2015.
• Shaw, W. T. and Buckley, I. R. C. The alchemy of probability distributions: Beyond Gram- Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map. Research report, 2007.
• Tahir, M. H., Cordeiro, G. M., Alizadeh, M. Mansoor, M. Zubair M. and Hamedani, G. G. The odd generalized exponential family of distributions with applications. Journal of Statistical Distributions and Applications, 2, 1-28, 2015.
• Xie, M. and Lai, C. D. Reliability analysis using an additive Weibull model with bathtub- shaped failure rate function. Reliability Engineering and System Safety, 52, 87-93, 1995.
• Yousof, H. M., Afify, A. Z., Alizadeh, M., Butt, N. S., Hamedani, G. G. and Ali, M. M. The transmuted exponentiated generalized-G family of distributions. Pakistan Journal of Statistics and Operation Research, 11, 441-464, 2015.
• Yousof, H. M., Afify, A. Z., Hamedani, G. G. and Aryal, G. (2016). The Burr X generator of distributions for lifetime data. Journal of Statistical Theory and Applications, forthcoming, 2016.
• Zografos K. and Balakrishnan, N. On families of beta- and generalized gamma generated distributions and associated inference. Statistical Methodology, 6, 344-362, 2009.