Main Article Content
Abstract
Transmutation is the functional composition of the cumulative distribution function (cdf) of one distribution with the inverse cumulative distribution function (quantile function) of another. Shaw and Buckley(2007), first apply this concept and introduced quadratic transmuted family of distributions. In this article, we have presented a review about the transmuted families of distributions. We have also listed the transmuted distributions, available in the literature along with some concluding remarks.
Keywords
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following License
CC BY: This license allows reusers to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use.
References
- Abayomi, A. (2019). Transmuted half normal distribution: Properties and application. Mathematical Theory and Modeling, 9:14-26.
- Abdul-Moniem, I. B. (2015). Transmuted Burr Type III Distribution. Journal of Statistics: Advances in Theory and Applications, 14:37-47.
- Abdul-Moniem, I. B. and Seham, M. (2015). Transmuted Gompertz Distribution. Computational and Applied Mathematics Journal, 1:88-96.
- Abdullahi, U. K. and Ieren, T. G. (2018). On the inferences and applications of transmuted exponential Lomax distribution. International Journal of Advanced Statistics and Probability, 6:30-36, doi:10.14419/ijasp.v6i1.8129.
- Afify, A. Z., Hamedani, G. G., Ghosh, I., and Mead, M. E. (2015a). The transmuted Marshall-Olkin Fre´chet distribution: Properties and applications. International Journal of Statistics and Probability, 4:132-148.
- Afify, A. Z., Nofal, Z. M., Yousof, H. M., El-Gebaly, Y. M., and Butt, N. S. (2015b). The transmuted Weibull-Lomax distribution: Properties and application. Pak.j.stat.oper.res., 11:135-152, doi:10.18187/pjsor.v11i1.956.
- Afify, A. Z., Yousof, H. M., Butt, N. S., and Hamedani, G. G. (2016). The transmuted Weibull-Pareto distri- bution. Pak. J. Statist., 32:183-206.
- Ahmad, A., Ahmad, S. P., and Ahmad, A. (2014). Transmuted inverse Rayleigh distribution: A generalization of the inverse Rayleigh distribution. Mathematical Theory and Modeling, 4:90-98.
- Ahmad, K., Ahmad, S. P., and Ahmed, A. (2015). Structural Properties of Transmuted Weibull Distribution. Journal of Modern Applied Statistical Methods, 14:141-158, doi:10.22237/jmasm/1446351120.
- AL-Kadim, K. A. (2018). Proposed Generalized Formula for Transmuted Distribution. Journal of University of Babylon, 26:66-74, doi:10.1515/eqc-2017-0027.
- AL-Kadim, K. A. and Mohammed, M. H. (2017). The cubic transmuted Weibull distribution. Journal of University of Babylon, 3:862-876.
- Alizadeh, M., Merovci, F., and Hamedani, G. G. (2017). Generalized transmuted family of distri- butions: Properties and applications. Hacettepe Journal of Mathematics and Statistics, 46:645-667, doi:10.15672/HJMS.201610915478.
- Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distribu- tions. METRON, 71:63-79, doi:10.1007/s40300-013-0007-y.
- Ansari, S. I. and Eledum, H. (2018). An Extension of Pareto Distribution. Journal of Statistics Applications & Probability, 7:443 455.
- Aryal, G. R. (2013). Transmuted Log-Logistic distribution. Journal of Statistics Applications & Probability, 2:11-20.
- Aryal, G. R. and Tsokos, C. P. (2009). On the transmuted extreme value distribution with application. Non- linear Analysis: Theory, Methods and Applications, 71:1401-1407, doi:10.1016/j.na.2009.01.168.
- Aryal, G. R. and Tsokos, C. P. (2011). Transmuted Weibull distribution: A generalization of the Weibull probability distribution. European Journal of Pure and Applied Mathematics, 4:89-102.
- Ashour, S. K. and Eltehiwy, M. A. (2013a). Transmuted exponentiated Lomax distribution. Australian Journal of Basic and Applied Sciences, 7:658-667.
- Ashour, S. K. and Eltehiwy, M. A. (2013b). Transmuted exponentiated modified Weibull distribution. Inter- national Journal of Basic and Applied Sciences, 2:258-269, doi:10.14419/ijbas.v2i3.1074.
- Ashour, S. K. and Eltehiwy, M. A. (2013c). Transmuted Lomax distribution. American Journal of Applied Mathematics and Statistics, 1:121-127, doi:10.12691/ajams-1-6-3.
- Aslam, M., Hussain, Z., and Asghar, Z. (2018). Cubic Transmuted-G Family of Distributions and Its Proper- ties. Stochastics and Quality Control, N/A:N/A, doi:10.1515/eqc-20170027.
- Balaswamy, S. (2018). Transmuted Half Normal Distribution. International Journal of Scientific Research in Mathematical and Statistical Sciences, 5:163-170.
- Bhatti, F. A., Hamedani, G. G., Sheng, W., and Ahmad, M. (2019). Cubic Rank Transmuted Modified Burr III Pareto Distribution: Development, Properties, Characterizations and Applications. International Journal of Statistics and Probability, 8:94-112, doi:10.5539/ijsp.v8n1p94.
- Bourguignon, M., Ghosh, I., and Cordeiro, G. M. (2016). General Results for the Transmuted Family of Distributions and New Models. Journal of Probability and Statistics, 2016:Article ID 7208425, 12 pages, doi:10.1155/2016/7208425.
- Bourguignon, M., LeaËœo, J., Leiva, V., and Santos-Neto, M. (2017). The transmuted birnbaum-saunders distribution. Revstat - Statistical Journal, 15:601-628.
- Burr, I. W. (1942). Cumulative frequency functions. Annals of Mathematical Statistics, 13:215-232, doi:10.1214/aoms/1177731607.
- Celik, N. (2018). Some Cubic Rank Transmuted Distributions. Journal of Applied Mathematics, Statistic and Informatics, 14:27-43.
- Chakraborty, S. and Bhati, D. (2016). Transmuted geometric distribution with applications in modelling and regression analysis of count data. Statistics and Operations Research Transactions, 40:153-176.
- Cordeiro, G. M. and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81:883-898, doi:10.1080/00949650903530745.
- Cordeiro, G. M., Saboor, A., Khan, M. N., Provost, S. B., and Ortega, E. M. M. (2017). The transmuted generalized modified weibull distribution. Filomat, 31:1395-1412, doi:10.2298/FIL1705395C.
- Das, K. K. (2015). On Some Generalised Transmuted Distributions. International Journal of Scientific & Engineering Research, 6:1686-1691.
- Elbatal, I. (2013a). Transmuted generalized inverted exponential distribution. Econ. Qual. Control, 28:125- 133, doi:10.1515/eqc-2013-0020.
- Elbatal, I. (2013b). Transmuted modified inverse Weibull distribution: A generalization of the modified inverse Weibull probability distribution. International Journal of Mathematical Archive, 4:117-119.
- Elbatal, I. and Aryal, G. (2013). On the transmuted additive Weibull distribution. Australian Journal of Statistics, 42:117-132, doi:10.17713/ajs.v42i2.160.
- Elbatal, I. and Aryal, G. (2015). Transmuted Dagum distribution with applications. Chilean Journal of Statistics, 6:31-45.
- Elbatal, I., Asha, G., and Raja, A. V. (2014). Transmuted exponentiated Fre´chet distribution: Properties and applications. Journal of Statistics Applications & Probability, 3:379-394, doi:10.12785/jsap/030309.
- Elbatal, I., Diab, L. S., and Alim, N. A. A. (2013). Transmuted generalized linear exponential distribution. International Journal of Computer Applications, 83:29-37.
- Elbatal, I. and Elgarhy, M. (2013). Transmuted quasi-Lindley distribution: A generalization of the quasi- Lindley distribution. Int. J. Pure Appl. Sci. Technol., 18:59-70.
- Elgarhy, M., Rashed, M., and Shawki, A. W. (2016). Transmuted generalized lindley distribution. In- ternational Journal of Mathematics Trends and Technology, 29:145-154, doi:10.14445/22315373/IJMTT- V29P520.
- Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31:497-512.
- Fatima, A. and Roohi, A. (2015). Transmuted exponentiated Pareto-i distribution. Pak. J. Statist., 32:63-80.
- Gharaibeh, M. M. and Al-Omari, A. I. (2019). Transmuted Ishita Distribution and Its Applications. Journal of Statistics Applications & Probability, 8:67-81.
- Granzotto, D. C. T. and Louzada, F. (2015). The Transmuted Log-Logistic Distribution: Modeling, Inference, and an Application to a Polled Tabapua Race Time up to First Calving Data. Communications in Statistics - Theory and Methods, 44:3387-3402, doi:10.1080/03610926.2013.775307.
- Granzotto, D. C. T., Louzada, F., and Balakrishnan, N. (2017). Cubic rank transmuted distributions: inferential issues and applications. Journal of Statistical Computation and Simulation, 87:2760-2778, doi.10.1080/00949655.2017.1344239.
- Gupta, R. C., Gupta, P., and Gupta, R. D. (1998). Modeling failure time data by Lehmann alternatives. Communications in Statistics-Theory and Methods, 27:887-904, doi:10.1080/03610929808832134.
- Haq, M. A., Butt, N. S., Usman, R. M., and Fattah, A. A. (2016). Transmuted power function distribution. Gazi University Journal of Science, 29:177-185.
- Hussian, M. A. (2014). Transmuted exponentiated gamma distribution: A generalization of exponentiated gamma probability distribution. Applied Mathematical Sciences,, 8:1297-1310, doi:10.12988/ams.2014.42105.
- Iriarte, Y. A. and Astorga, J. M. (2014). Transmuted Maxwell probability distribution. Revista Integracio´n, 32:211-221.
- Iriarte, Y. A. and Astorga, J. M. (2015). A version of transmuted generalized Rayleigh distribution. Revista Integracio´n, 33:83-95.
- Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1. John Wiley & Sons, New York, USA.
- Johnson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2. John Wiley & Sons, New York, USA.
- Khan, M. and King, R. (2013). Transmuted modified Weibull distribution: A generalization of the modified Weibull probability distribution. European Journal of Pure and Applied Mathematics., 6:66-88.
- Khan, M. S. and King, R. (2014a). A New Class of Transmuted Inverse Weibull Distribution for Reliability Analysis. American Journal of Mathematical and Management Sciences, 33:261-286, doi:10.1080/01966324.2014.929989.
- Khan, M. S. and King, R. (2014b). Transmuted generalized inverse Weibull distribution. Journal of Applied Statistical Science, 20:213-230.
- Khan, M. S. and King, R. (2015). Transmuted Modified Inverse Rayleigh Distribution. Austrian Journal of Statistics, 44:17-29, doi:10.17713/ajs.v44i3.21.
- Khan, M. S., King, R., and Hudson, I. L. (2014). Characterizations of the transmuted inverse Weibull distri- bution. ANZIAM J., 55:C197-C217.
- Khan, M. S., King, R., and Hudson, I. L. (2015). A new three parameter transmuted Chen lifetime distribution with application. Journal of Applied Statistical Sciences, 21:239-259.
- Khan, M. S., King, R., and Hudson, I. L. (2016a). Transmuted Gompertz distribution: Properties and estima- tion. Pak. J. Statist., 32:161-182.
- Khan, M. S., King, R., and Hudson, I. L. (2016b). Transmuted Kumaraswamy distribution. Statistics in Transition, 17:1-28, doi:10.21307/stattrans-2016-013.
- Khan, M. S., King, R., and Hudson, I. L. (2017a). Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data. Communications in Statistics- Simulation and Computation, 46:4377-4398 , doi:10.1080/03610918.2015.1118503.
- Khan, M. S., King, R., and Hudson, I. L. (2017b). Transmuted new generalized inverse Weibull distribution. Pak.j.stat.oper.res., 13:277-296, doi:10.18187/pjsor.v13i2.1523.
- Khan, M. S., King, R., and Hudson, I. L. (2017c). Transmuted Weibull distribution: Properties and estimation. Communications in Statistics - Theory and Methods, 46:5394-5418, doi:10.1080/03610926.2015.1100744.
- Khan, M. S., King, R., and Hudson, I. L. (2018). Transmuted Modified Weibull distribution: Properties and Application. European Journal of Pure and Applied Mathematics, 11:362-374.
- Khan, M. S., King, R., and Hudson, I. L. (2019). Transmuted Burr Type X Distribution with Covariates Regression Modeling to Analyze Reliability Data. American Journal of Mathematical and Management Sciences, doi:10.1080/01966324.2019.1605320.
- Kumaraswamy, P. (1980). A Generalized probability density-function for double-bounded random-processes. Journal of Hydrology, 462:79-88.
- Lucena, S. E. F., Silva, A. H. A., and Cordeiro, G. M. (2015). The transmuted generalized gamma distribution: Properties and application. Journal of Data Science, 13:409-420.
- Luguterah, A. and Nasiru, S. (2015). Transmuted exponential Pareto distribution. Far East Journal of Theoretical Statistics, 50:31-49.
- Mahmoud, M. R. and Mandouh, R. M. (2013). On the transmuted fre´chet distribution. Journal of Applied Sciences Research, 9:5553-5561.
- Mansour, M. M., Elrazik, E. M. B., Hamed, M. S., and Mohamed, S. M. (2015). A new transmuted additive Weibull distribution: Based on a new method for adding a parameter to a family of distribution. International Journal of Applied Mathematical Sciences, 8:31-54.
- Mansour, M. M. and Mohamed, S. M. (2015). A new generalized of transmuted Lindley distribution. Applied Mathematical Sciences, 9:2729-2748, doi:10.12988/ams.2015.52158.
- Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84:641-652, doi:10.1093/biomet/84.3.641.
- Merovci, F. (2013a). Transmuted exponentiated exponential distribution. Mathematical Sciences And Appli- cations E-Notes, 1:112-122.
- Merovci, F. (2013b). Transmuted Lindley distribution. Int. J. Open Problems Comput. Math., 6:63-72.
- Merovci, F. (2014). Transmuted Generalized Rayleigh Distribution. Journal of Statistics Applications & Probability, 3:9-20, doi:10.18576/jsap/030102.
- Merovci, F., Alizadeh, M., and Hamedani, G. G. (2016). Another Generalized Transmuted Family of Distri- butions: Properties and Applications. Austrian Journal of Statistics, 45:71-93, doi:10.17713/ajs.v45i3.109.
- Merovci, F., Elbatal, I., and Ahmed, A. (2014). Transmuted generalized inverse Weibull distribution. Aus- tralian Journal of Statistics, 43:119-131.
- Merovci, F. and Puka, L. (2014). Transmuted Pareto Distribution. ProbStat Forum, 7:1-11.
- Nofal, Z. M., Afify, A. Z., Yousof, H. M., Granzotto, D. C. T., and Louzada, F. (2018). The Transmuted Expo- nentiated Additive Weibull Distribution: Properties and Applications. Journal of Modern Applied Statistical Methods, 17(1), eP2526. doi:10.22237/jmasm/1525133340.
- Okorie, I. E. and Akpanta, A. C. (2019). A Note on the Transmuted Generalized Inverted Exponential Distribution with Application to Reliability Data. Thailand Statistician, 17:118-124.
- Otiniano, C. E. G., de Paiva, B. S., Daniele, S. B., and Neto, M. (2019). The transmuted generalized extreme value distribution: properties and application. Communications for Statistical Applications and Methods, 26:239-259.
- Owoloko, E. A., Oguntunde, P. E., and Adejumo, A. O. (2015). Performance rating of the transmuted expo- nential distribution: An analytical approach. Springer Plus, 4:8-18, doi:10.1186/s40064-015-1590-6.
- Pearson, K. (1895). Contributions to the mathematical theory of evolution, ii: Skew variation in homogeneous material. Philosophical Transactions of the Royal Society, 186:343-414, doi:10.1098/rsta.1895.0010.
- Pearson, K. (1901). Mathematical contributions to the theory of evolution, x: Supplement to a memoir on skew variation. Philosophical Transactions of the Royal Society, 197:443 459, doi:10.1098/rsta.1901.0023.
- Pearson, K. (1916). Mathematical contributions to the theory of evolution, xix: Second supple- ment to a memoir on skew variation. Philosophical Transactions of the Royal Society, 216:429-457, doi:10.1098/rsta.1916.0009.
- Pobocˇ´ıkova´, I., Sedliacˇkova´, Z., and Michalkova´, M. (2018). Transmuted Weibull distribution and its appli- cations. MATEC Web of Conferences, 157:1-11, doi:10.1051/matecconf/201815708007.
- Rahman, M. M., Al-Zahrani, B., and Shahbaz, M. Q. (2018a). A General Transmuted Family of Distributions. Pak.j.stat.oper.res., 14:451-469, doi:10.18187/pjsor.v14i2.2334.
- Rahman, M. M., Al-Zahrani, B., and Shahbaz, M. Q. (2018b). Cubic Transmuted Pareto Distribution. Annals of Data Science, doi:10.1007/s40745-018-0178-8.
- Rahman, M. M., Al-Zahrani, B., and Shahbaz, M. Q. (2018c). New General Transmuted Family of Distribu- tions with Applications. Pak J Stat Oper Res, 14:807-829, doi:10.18187/pjsor.v14i4.2655.
- Rahman, M. M., Al-Zahrani, B., and Shahbaz, M. Q. (2019a). Cubic Transmuted Weibull Distribution: Properties and Applications. Annals of Data Science.
- Rahman, M. M., Al-Zahrani, B., Shahbaz, S. H., and Shahbaz, M. Q. (2019b). Cubic Transmuted Uniform Distribution: An Alternative to Beta and Kumaraswamy Distributions. European Journal of Pure and Applied Mathematics, 12:1106-1121.
- Samuel, A. F. (2019). On the Performance of Transmuted Logistic Distribution: Statistical Properties and Application. Budapest International Research in Exact Sciences (BirEx) Journal, 1:26-34.
- Shahzad, M. N. and Asghar, Z. (2016). Transmuted Dagum distribution: A more flexible and broad shaped hazard function model. Hacettepe Journal of Mathematics and Statistics, 45:227,244.
- Shaw, W. T. and Buckley, I. R. C. (2007). The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. Research report.
- Tadikamalla and Pandu, R. (1980). A Look at the Burr and Related Distributions. International Statistical Review, 48:337-344.
- Tahir, M. H. and Cordeiro, G. M. (2016). Compounding of distributions: a survey and new generalized classes. Journal of Statistical Distributions and Applications, 3:1-35, doi:10.1186/s40488-016-0052-1.
- Tian, Y., Tian, M., and Zhu, Q. (2014). Transmuted linear exponential distribution: A new generalization of the linear exponential distribution. Communications in Statistics - Simulation and Computation, 43:2661-2671, doi:10.1080/03610918.2013.763978.
- Vardhan, R. V. and Balaswamy, S. (2016). Transmuted new modified Weibull distribution. Mathematical Sciences and Applications E-Notes, 4:125-135.