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Statistics in Transition New Series

Polish Statistical Association

Central Statistical Office of Poland

Subject: Economics, Statistics & Probability


ISSN: 1234-7655
eISSN: 2450-0291





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VOLUME 17 , ISSUE 2 (June 2016) > List of articles


Muhammad Shuaib Khan * / Robert King * / Irene Lena Hudson *

Keywords : Kumaraswamy distribution, moments, order statistics, parameter estimation, maximum likelihood estimation

Citation Information : Statistics in Transition New Series. Volume 17, Issue 2, Pages 183-210, DOI:

License : (CC BY 4.0)

Published Online: 06-July-2017



The Kumaraswamy distribution is the most widely applied statistical distribution in hydrological problems and many natural phenomena. We propose a generalization of the Kumaraswamy distribution referred to as the transmuted Kumaraswamy (𝑇𝐾𝑀) distribution. The new transmuted distribution is developed using the quadratic rank transmutation map studied by Shaw et al. (2009). A comprehensive account of the mathematical properties of the new distribution is provided. Explicit expressions are derived for the moments, moment generating function, entropy, mean deviation, Bonferroni and Lorenz curves, and formulated moments for order statistics. The 𝑇𝐾𝑀 distribution parameters are estimated by using the method of maximum likelihood. Monte Carlo simulation is performed in order to investigate the performance of MLEs. The flood data and HIV/ AIDS data applications illustrate the usefulness of the proposed model.

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  1. ASHOUR, S. K., ELTEHIWY, M. A., (2013). Transmuted Lomax Distribution, American Journal of Applied Mathematics and Statistics, 1(6), pp. 121βˆ’127.
  2. AHMAD, A., AHMAD, S. P., AHMED, A., (2015). Characterization and estimation of transmuted Kumaraswamy distribution, Mathematical Theory and Modeling, Vol. 5, No. 9, 168βˆ’174.
  3. ARYAL, G. R., TSOKOS, C. P., (2011). Transmuted Weibull distribution: A Generalization of the Weibull Probability Distribution. European Journal of Pure and Applied Mathematics, Vol. 4, No. 2, 89βˆ’102.
  4. ARYAL, G. R., TSOKOS. C. P., (2009). On the transmuted extreme value distribution with applications. Nonlinear Analysis: Theory, Methods and applications, Vol. 71, 1401βˆ’1407.
  5. ARYAL, G. R., (2013). Transmuted Log-Logistic Distribution, J. Stat. Appl. Pro. 2, No. 1, 11βˆ’20.
  6. BALAKRISHNAN, A. N., NAGARAJA, H. N., (1992). A first course in order statistics. New York: Wiley-Interscience.
  7. CORDEIRO, G. M., ORTEGA, E. M. M., NADARAJAH, S., (2010). The Kumaraswamy Weibull distribution with application to failure data. J Frankl Inst 347, 1399–1429.
  8. CORDEIRO, G. M., NADARAJAH, S., ORTEGA, E. M. M., (2012). The Kumaraswamy Gumbel distribution, Stat Methods Appl 21: 139–168.
  9. CORDEIRO G. M., GOMES, A. E., QUEIROZ DA-SILVA, C., ORTEGA, E. M. M., (2013). The beta exponentiated Weibull distribution, Journal of Statistical Computation and Simulation. 83, 1, 114–138.
  10. CORDEIRO, G. M., DE CASTRO M., (2009). A new family of generalized distributions, Journal of Statistical Computation & Simulation, Vol. 00, No. 00, August, 1–17.
  11. DUMONCEAUX, R., ANTLE, C. E., (1973). Discrimination between the log-normal and the Weibull distributions. Technometrics 15 (4), 923–926.
  12. ELBATHAL, I., ELGARHY, M., (2013). Transmuted Quasi Lindley Distributions: A generalization of the Quasi Lindley Distribution. International Journal of Pure and Applied Sciences and Technology , 18(2), pp. 59–69.
  13. FLETCHER, S. C., PONNAMBALAM, K., (1996). Estimation of reservoir yield and storage distribution using moments analysis. J Hydrol 182, 259–275.
  14. GANJI, A., PONNAMBALAM, K., KHALILI, D., (2006). Grain yield reliability analysis with crop water demand uncertainty. Stoch Environ Res Risk Assess 20, 259–277.
  15. JONES, M. C., (2009). Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. Stat Methodol 6, 70–91.
  16. KENNEY, J. F., KEEPING, E. S., (1962). Mathematics of Statistics. Princeton, NJ.
  17. KUMARASWAMY, P., (1980). Generalized probability density-function for double-bounded random-processes. J Hydrol 46, 79–88.
  18. KHAN, M. S., KING, R., (2013a). Transmuted Modified Weibull Distribution: A Generalization of the Modified Weibull Probability Distribution, European Journal of Pure And Applied Mathematics, Vol. 6, No. 1, 66–88.
  19. KHAN, M. S., KING, R., (2013b). Transmuted generalized Inverse Weibull distribution, Journal of Applied Statistical Sciences, Vol. 20, No. 3, 15–32.
  20. KHAN, M. S., KING, R., HUDSON, I., (2013c). Transmuted generalized Exponential distribution, 57th Annual Meeting of the Australian Mathematical Society, Australia.
  21. KHAN, M., SHUAIB, KING, R., HUDSON, I., (2014). Characterizations of the transmuted Inverse Weibull distribution, ANZIAM J., Vol. 55 (EMAC2013 at the Queensland University of Technology from 1st – 4th December 2013), C197–C217.
  22. KOUTSOYIANNIS, D, XANTHOPOULOS, T., (1989). On the parametric approach to unit hydrograph identification. Water Resour Manag 3, 107–128.
  23. MOORS, J. A., (1998). A quantile alternative for kurtosis. Journal of the Royal Statistical Society, D, 37, 25–32.
  24. MEROVCI, F., (2013a). Transmuted Rayleigh distribution. Austrian Journal of Statistics, Vol. 42, No. 1, 21–31.
  25. MEROVCI, F., (2013b). Trasmuted Lindley distribution. Int. J. Open Problems Compt. Math, 6, 63–72.
  26. MEROVCI, F., (2014). Transmuted generalized Rayleigh distribution, J. Stat. Appl. Pro. 3, No. 1, 9–20.
  27. NADARAJAH, S., (2008). On the distribution of Kumaraswamy. J Hydrol 348, 568–569.
  28. PRINCIPE, J. C., (2009).
  29. PONNAMBALAM, K, SEIFI, A, VLACH, J., (2001). Probabilistic design of systems with general distributions of parameters. Int J Circuit Theory Appl 29, 527–536.
  30. RΓ‰NYI, A., (1961). On measures of entropy and information. University of California Press, Berkeley, California, 547–561.
  31. SUNDAR, V., SUBBIAH, K., (1989). Application of double bounded probability density-function for analysis of ocean waves. Ocean Eng 16, 193–200.
  32. SEIFI, A., PONNAMBALAM, K., VLACH, J., (2000). Maximization of manufacturing yield of systems with arbitrary distributions of component values. Ann Oper Res 99, 373–383.
  33. SHAW, W. T., BUCKLEY, I. R. C., (2009). The alchemy of probability distributions: beyond Gram–Charlier expansions, and a skew-kurtotic normal distribution from a rank transmutation map. Technical report,