Mixed Topp-Leone-Kumaraswamy distribution

Document Type : Research Paper


Department of Information Technology, Technical College of Management-Baghdad, Middle Technical University, Baghdad, Iraq


In this article, a new generalization of the Topp-Leone distribution with a unit interval, namely Mixed Topp-Leone-Kumaraswamy distribution is defined and studied. The mathematical properties of this mixing distribution are described. Moments, quantile function, R?nyi entropy, incomplete moments and moments of residual are obtained for the new Mixed Topp-Leone - Kumaraswamy distribution. The maximum likelihood (MLE), Crans (CM) , Percentile (PM) and Particle Swarm Optimization(PSO) estimators of the parameters are derived. The percentile Method is more efficient method as compred to the others. Two real data sets are used to illustrate an application and superiority of the proposed distribution.


[1] A. Alzaatreh, C. Lee and F. Famoye, A new method for generating families of continuous distributions , Metron. 71(1) (2013) 63–79.
[2] G. R. Aryal and C. P. Tsokos, Transmuted Weibull distribution: A generalization of the Weibull probability distribution, European J. Pure Appl. Math. 4(2) (2011) 89–102.
[3] S.H. Bhatti, S. Hussain, T. Ahmad, M. Aslam, M. Aftab and M.A. Raza, Efficient estimation of Pareto model: Some modified percentile estimators, PLoS ONE 13(5) (2018).
[4] G.M. Cordeiroa and M. Castro, A new family of generalized distributions, J. Stat. Comput. Sim. (2010) 1563–5163.
[5] G.W. Cran, Moment estimators for the 3-parameter Weibull distribution, IEEE Trans. Rel. 37(4) (1988).
[6] S.D. Dubey, Some percentile estimators for Weibull parameters, Techno. 9 (1967) 119–129.
[7] D. A. Gupta, Asymptotic Theory of Statistics and Probability, Science Business Media, LLC, 2008.
[8] M. Elgarhy, M.A.U. Haq and Q.U. Ain, Exponentiated generalized Kumaraswamy distribution with applications, Ann. Data Sci. 5(2) (2018) 273–292.
[9] M. M. Hasan, B. F. W. Croke, S. Liu, K. Shimizu and F. Karim, Using mixed probability distribution functions for modeling non-zero sub-daily rainfall in Australia, Ann. Data Sci. Geos. 10 (43) (2020).
[10] S. Hashmi, M. A. U.Haq and R.M. Usman, A generalized exponential distribution with increasing, decreasing and constant shape hazard curves, Elec. J. Appl. Stat. Analy. 12(1) (2019) 223–244.
[11] M.A.U. Haq, M. Elgarhy, S. Hashmi, G. Ozel and Q.U. Ain, Transmuted Weibull power function distribution: its properties and applications, J. Data Sci. 397 (2018) 418.
[12] I.R. James, Estimation of the mixing proportion in a mixture of two normal distributions from simple, rapid measurements, Biomet. 34(2) (1978) 265–275.
[13] M. C. Jones, Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages, Stat. Meth. 6 (2009) 70–81.
[14] P. Kumaraswamy, Generalized probability density-function for double- bounded random processes, J. Hydro. 46 (1980) 79–88.
[15] J. Kennedy J and R. Eberhart, Particle swarm optimization, Int. Conf. Neural Networks; Australia: Perth; 1995. 1942–1948.
[16] A.J. Lemontea, W.B. Souzaa and G. M. Cordeirob, The exponentiated Kumaraswamy distribution and its logtransform, Brazilian J. Prob. Stat. 27(1) (2013) 31–53.
[17] Y.M. Mehmet Yilmaz and B. Buyum, Parameter estimation methods for two-component mixed exponential distributions, J. Turkish Stat. Assoc. 8(3) (2015) 51–59.
[18] Y.A. Mohammed, B. Yatim and S. Ismail, A parametric mixture model of three different distributions: An approach to analyze heterogeneous survival data, AIP Conf. Proc. 1605, 1040; (2014).
[19] M.J. Mohammed and A.T. Mohammed, Parameter estimation of inverse exponential Rayleigh distribution based on classical methods, Int. J. Nonlinear Anal. Appl. 12(1) (2021) 935–944 .
[20] S. Nadarajah, and S. Kotz, Moments of some J-shaped distribution, J. Appl. Stat. 30 (2003) 311–317.
[21] A. M. Nigm, E. K. AL-Hussaini, and Z. F. Jaheen, Bayesian one- sample prediction of future observations under Pareto distribution, Stat. 37(6) (2003) 527–536.
[22] R. Silva, F. Gomes-Silva, M. Ramos, G. Cordeiro, P. Marinho and T.A.N. De Andrade , The exponentiated Kumaraswamy-G class: general properties and application, Revista Colombiana Estad. 42(1) (2019) 1–33.
[23] R. B. Silva, M. Bourguignon, C.R.B. Dias and G.M. Cordeiro, The compound class of extended Weibull power series distributions, Comput. Stat. Data Anal. 58 (2013) 352–367.
[24] W. Szulczewski and W. Jakubowski, The application of mixture distribution for the estimation of extreme floods in controlled catchment basins, Water Resource Manag. 32 (2018) 3519–3534.
[25] C.W. Topp and F.C. Leone, A Family of J-shaped frequency functions , J. Amer. Stat. Assoc. 50 (1955) 209–219.
[26] R.A. Zeineldin, M. Ahsan Ul Haq, S. Hashmi and M. Elsehety, Alpha power transformed inverse Lomax distribution with different methods of estimation and applications, Complexity. 2020 Article ID (1860813) (2020).
[27] X. Zhai, J. Wang and J. Chen, Parameter estimation method of mixture distribution for construction machinery, Math. Prob. Engin. 2018, Article ID 3124048 (2018).
[28] G. Q. Zhang, Parameters estimation of three mixed exponential distributions, Int. Conf. Elect. Autom. Mech. Engin. (EAME 2015).
Volume 12, Issue 2
November 2021
Pages 699-715
  • Receive Date: 11 March 2021
  • Revise Date: 20 April 2021
  • Accept Date: 25 May 2021