On a new approach to distributions with variable transmuting parameter: The concept and examples with emerging problems

Year 2022, Volume: 2 Issue: 2, 73 - 87, 30.06.2022

Jordan Hristov

https://doi.org/10.53391/mmnsa.2022.007

Cited By: 1

Abstract

A new concept in the transmutation of distribution applying variable transmuting function has been conceived. Test examples with power function by quadratic and cubic transmutations have been demonstrated by the applications of the error-function and standard logistic function variable transmuting functions. The efficiency and properties of the new approach by numerical examples addressing the rate constants of the transmuting functions and the shape parameter of the test power function have been demonstrated. An additional example with a quadratic transmutation of the exponential distribution through the error function as a variable transmuting parameter has been developed.

Keywords

Transmutation, variable transmuting parameter, transmuted distributions, power function

References

• Dey, S., Kumar, D., Anis, M.Z., Nadarajah, S., & Okorie, I. A review of transmuted distributions. Journal of the Indian Society for Probability and Statistics, 22(1), 47-111, (2021).
• Cordeiro, G.M., Ortega, E.M.M., & Popovic, B.V. The gamma-Lomax distribution. Journal of Statistical computation and Simulation, 85(2), 305-319, (2015).
• Tahir, M.H., & Cordeiro, G.M. Compounding of distributions : a survey and new generalized classes. Journal of Statistical Distributions and Applications, 3(1) 1-35, (2016).
• Afuecheta, E., Semeyutin, A., Chan, S., Nadarajah, S., & Ruiz, D.A.P. Compound distributions for financial returns. Plos One, 15(10), e0239652.
• Al-Hussaini, E.K., & Ahsanullah, M. Exponentiated distributions. Atlantis Studies in Probability and Statistics, 21, (2015).
• Alexander, C. Cordeiro, G.M., Ortega, E.M., & Sarabia, J.M. Generalized beta-generated distributions. Computational Statistics & Data Analysis, 56(6), 1880-1897, (2012).
• Alizadeh, M., Cordeiro, G.M., Brito, E.D., & Demetrio, C.G.B. The beta Marshal-Olkin family of distributions. Journal of Statistical Distributions and Applications, 2(1), 1-18, (2015).
• Marshall, A.W., & Olkin, I. 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, (1997).
• Patil, G.P., & Rao, C.R. Weighted distributions and size-biased sampling with applications to wildlife populations and human families. Biometrics, 34(2), 179-189, (1978).
• Bakouch, H., Chesneau, C., & Enany, M. A weighted general family of distributions: Theory and practice. Computational and Mathematical Methods, 3(6), e1135, (2021).
• Shaw W., & Buckley, I.R. The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtosis normal distribution from a rank transmutation map. Research Report. UCL discovery repository, arXiv preprint: 0901.0434, Statistical Finance), (2009).
• Almalki, S.J., & Nadarajh, S. Modification of the Weibull distribution: a review. Reliability Engineering & System Safety, 124, 32-55, (2014).
• Lai, M.T. Optimum number of minimal repairs for a system under increasing failure rate shock model with cumulative repair-cost limit. International Journal of Reliability and Safety, 7(2), 95-107, (2013).
• Elbatal, I. Transmuted modified inverse Weibull distribution: a generalization of the modified inverse Weibull probability distribution. International Journal of Mathematical Archive, 4(8), 117-129, (2013).
• Afify, A.Z., Nofal, Z.M., Yousuf, H.M., El Gebaly, Y.M., & Butt, N.S. The transmuted Weibull Lomax distribution: properties and application. Pakistan Journal of Statistics and Operation Research, 11, 135-152, (2015).
• Granzotto, D.C.T., Louzada, F. & Balakrishnan N. Cubic rank transmuted distributions: inferential issues and applications. Journal of statistical Computation and Simulation, 87(14), 2760–2778, (2017).
• Khan, M.S., & King,R. Transmuted modified Weibul distribution: a generalization of the modified Weibull probability distribution. European Journal of pure and applied mathematics, 6(1), 66-88, (2013).
• Okorie, I.E., Akpanta, A.C., Ohakwe, J., & Chikezie, D.C., The modified Power function distribution. Cogent Mathematics, 2017(4), 1219592, (2017).
• Sakthivel, K.M., Rajitha, C.S., & Dhivakar, K. Two parameter cubic rank transmutation of Lindley distribution. AIP Conference Proceedings , 2261, 030086, (2020).
• Rahman, M.M., Al-Zahrani, B., & Shahbaz, M.Q. A general transmuted family of distributions. Pakistan Journal of Statistics and Operation Research, 14(2), 451-469, (2018).
• Ghosh, S., Kataria, K.K., & Vellaisamy, P. On transmuted generalized linear exponential distribution. Communications in Statistics - Theory and Methods, 50(9), 1978-2000, (2021).
• Rahman, M.M., Al-Zahrani, B., Shahbaz, S.H., & Shahbaz, M.Q. Transmuted probability distributions: A review. Pakistan Journal of Statistics and Operation Research, 16(1), 83-94, (2020).
• Celik, N. Some cubic rank transmuted distributions. Journal of Applied Mathematics, Statistics and Informatics, 14(2), 27-43, (2018).
• Tian, Y., Tian, M., Zhu, Q. Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution. Communications in Statistics-Simulation and Computation, 43(10), 2661-2677, (2014).
• Alizadeh, M., Merovci, F., & Hamedani, G.G. Generalized transmuted family of distributions: properties and applications. Hacettepe Journal of Mathematics and Statistics, 46(4), 645-667, (2017).
• Merovci, F., Alizadeh, M., & Hamedani, G.G. Another generalized transmuted family of distributions: properties and applications. Austrian Journal of Statistics, 45(3), 71-93, (2016).
• Bourguignon, M., Gosh, I., & Cordeiro, G.M. General; results for transmuted family of distributions and new models. Journal of Probability and Statistics, (2016). ID: 7208425.
• Aryal, G.R., & Tsokos, C.P. On the transmuted extreme value distribution with application. Nonlinear Analysis: Theory, Methods & Applications, 71(12), e1401-e1407, (2009).
• Aryal, G.R., & Tsokos, C.P. Transmuted Weibull distribution: A generalized of the Weibull probability distribution. European Journal of pure and applied mathematics, 4(2), 89-102, (2011).
• Koopmans, L.H. Some simple singular and mixed probability distributions. Some simple singular and mixed probability distributions, 76(3), 297-299, (1969).
• Karlis. D., & Xekalaki,E. Mixed poisson distributions. International Statistical Review/Revue Internationale de Statistique, 73(1), 35-58, (2005).
• Fischer, S., Schumann, A., & Schulte, M. Characterisation of seasonal flood types according to timescales in mixed probability distributions. Journal of Hydrology, 539, 38-56, (2016).
• McLachlan, G.J., & Peel, D. Finite mixture models. John Wiley & Sons, (2004).
• Owoloko, E.A., Oguntunde, P.E., & Adejumo, A.O. Performance rating of the transmuted exponential distribution: An analytical approach. Springer Plus, 4(1), 1-15, (2015).
• Hristov, J. Integral-balance method with transmuted profiles: Concept, examples and emerging problems. Journal of Computational and Applied Mathematics, in press, (2022).