The Kumaraswamy Distribution: Statistical Properties and Application

Duraid Hussein Badra, Alyaa Hashem Mohammed


Modeling and analyzing lifetime data is an important aspect of statistical work in various scientific and technological fields such as medicine, engineering, insurance, and finance. The modeling and analysis of lifetimes is an important aspect of statistical work in various scientific and technological fields. In recent years, inverted Kumaraswamy distribution has been used quite effectively to model many lifetime data. The most broadly applied statistical distribution is Kumaraswamy distribution in hydrological problems and many natural phenomena. The Kumaraswamy distribution (KD) is widely applied for modeling data in practical domains, such as medicine, engineering, economics, and physics. The present work proposes the Bayesian estimators of KD parameters through the use of type-II censoring data in this research the problem to estimate the unknown parameters of Kumaraswamy distribution with two parameters θ and λ, these estimates are a maximum likelihood of ordered observation and the Bayesian for the parameter of the Kumaraswamy distribution (KUD) depended on ranked set sampling (RSS) techniques. Both the simulated are inserted into real-life data sets and are considered to make a comparison between the estimation based on Maximum Likelihood estimators and Bayesian Estimation methods based on (RSS) techniques. For comparison purposes, we employed (100) mean square error and the criteria like AICC (Akaike information corrected criterion). Finally, the importance and flexibility of the new model of real data set are proved empirically.


Kumaraswamy distribution; Bayes estimation; reliability analysis; failure function; quantile function; order statistics.

Full Text:



P. Kumaraswamy, “A generalized probability density function for double-bounded random processes,†J. Hydrol., vol. 46, no. 1–2, pp. 79–88, 1980.

V. A. Gonzalez-Lopez, R. Gholizadeh, and C. E. Galarza, “E-Bayesian estimation for system reliability and availability analysis based on exponential distribution,†Commun. Stat. Comput., vol. 46, no. 8, pp. 6221–6241, 2017.

M. A. Hussian, “Bayesian and maximum likelihood estimation for Kumaraswamy distribution based on ranked set sampling,†Am. J. Math. Stat., vol. 4, no. 1, pp. 30–37, 2014.

N. Feroze and I. El-Batal, “Parameter estimations based on Kumaraswamy progressive type II censored data with random removals,†J. Mod. Appl. Stat. Methods, vol. 12, no. 2, p. 19, 2013.

M. S. Khan, R. King, and I. L. Hudson, “Transmuted kumaraswamy distribution,†Stat. Transit. new Ser., vol. 17, no. 2, pp. 183–210, 2016.

S. Hashmi, K. Aidi, P. L. Ramos, and F. Louzada, “Unit modified Burr-III distribution: Estimation, characterizations and validation test,†Ann. Data Sci., pp. 1–26, 2020.

M. M. Eldin, N. Khalil, and M. Amein, “Estimation of parameters of the Kumaraswamy distribution based on general progressive type II censoring,†Am. J. Theor. Appl. Stat., vol. 3, no. 6, pp. 217–222, 2014.

M. Ç. Korkmaz, C. Chesneau, and Z. S. Korkmaz, “On the arcsecant hyperbolic normal distribution. Properties, quantile regression modeling and applications,†Symmetry (Basel)., vol. 13, no. 1, p. 117, 2021.

R. A. R. Bantan, C. Chesneau, F. Jamal, M. Elgarhy, W. Almutiry, and A. A. Alahmadi, “Study of a Modified Kumaraswamy Distribution,†Mathematics, vol. 9, no. 21, p. 2836, 2021.

M. Garg, “On distribution of order statistics from Kumaraswamy distribution,†Kyungpook Math. J., vol. 48, no. 3, pp. 411–417, 2008.

H. M. Okasha, “E-Bayesian estimation for the exponential model based on record statistics,†J. Stat. Theory Appl., vol. 18, no. 3, pp. 236–243, 2019.

H. M. Reyad and S. O. Ahmed, “Bayesian and E-Bayesian estimation for the Kumaraswamy distribution based on type-II censoring,†Int. J. Adv. Math. Sci., vol. 4, no. 1, pp. 10–17, 2016.

T. Kayal, Y. M. Tripathi, D. Kundu, and M. K. Rastogi, “Statistical inference of Chen distribution based on type I progressive hybrid censored Samples,†Stat. Optim. Inf. Comput., vol. 10, no. 2, pp. 627–642, 2022.

A. Algarni, A. M. Almarashi, H. Okasha, and H. K. T. Ng, “E-bayesian estimation of chen distribution based on type-I censoring scheme,†Entropy, vol. 22, no. 6, p. 636, 2020.

U. Kamps, “A concept of generalized order statistics,†J. Stat. Plan. Inference, vol. 48, no. 1, pp. 1–23, 1995.

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,†Rev. Colomb. Estadística, vol. 42, no. 1, pp. 1–33, 2019.

A. Bekker, J. J. J. Roux, and P. J. Mosteit, “A generalization of the compound Rayleigh distribution: using a Bayesian method on cancer survival times,†Commun. Stat. Methods, vol. 29, no. 7, pp. 1419–1433, 2000.

C. Kuş, “A new lifetime distribution,†Comput. Stat. Data Anal., vol. 51, no. 9, pp. 4497–4509, 2007.

B. E. Mohammed, “Statistical properties of Kumaraswamy-generalized exponentiated exponential distribution,†Int. J. Comput. Appl., vol. 94, no. 4, pp. 1–8, 2014.

R. M. El-Sagheer, “Estimating the parameters of Kumaraswamy distribution using progressively censored data,†J. Test. Eval., vol. 47, no. 2, pp. 905–926, 2019.

M. Sagrillo, R. R. Guerra, and F. M. Bayer, “Modified Kumaraswamy distributions for double bounded hydro-environmental data,†J. Hydrol., vol. 603, p. 127021, 2021.

T. N. Sindhu, N. Feroze, and M. Aslam, “Bayesian analysis of the Kumaraswamy distribution under failure censoring sampling scheme,†Int. J. Adv. Sci. Technol., vol. 51, pp. 39–58, 2013.

F. S. Alduais, M. F. Yassen, M. M. A. Almazah, and Z. Khan, “Estimation of the Kumaraswamy distribution parameters using the E-Bayesian method,†Alexandria Eng. J., vol. 61, no. 12, pp. 11099–11110, 2022.

U. Kamps, “A Concept of Generalized Order Statistics. Teubner, Stuttgart,†Update, vol. 3, pp. 553–557, 1995.

S. Yue, P. Pilon, and G. Cavadias, “Power of the Mann–Kendall and Spearman’s rho tests for detecting monotonic trends in hydrological series,†J. Hydrol., vol. 259, no. 1–4, pp. 254–271, 2002.

N. J. Al-Anber, “Mixed Topp-Leone-Kumaraswamy distribution,†Int. J. Nonlinear Anal. Appl., vol. 12, no. 2, pp. 699–715, 2021.

J. Varghese and J. KK, “Kumaraswamy harris generalized kumaraswamy distribution and its Application in Survival Analysis,†Biom Biostat Int J, vol. 11, no. 1, pp. 28–34, 2022.

H. Piriaei, G. Yari, and R. Farnoosh, “On E-Bayesian estimations for the cumulative hazard rate and mean residual life under generalized inverted exponential distribution and type-II censoring,†J. Appl. Stat., vol. 47, no. 5, pp. 865–889, 2020.

A. Kohansal, “On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample,†Stat. Pap., vol. 60, no. 6, pp. 2185–2224, 2019.

P. K. Vishwakarma and P. Dutta, “H i column density statistics of the cold neutral medium from absorption studies,†Mon. Not. R. Astron. Soc., vol. 491, no. 2, pp. 2360–2365, 2020.

P. Kumaraswamy, “Extended sinepower probability density function,†J. Hydrol., vol. 37, no. 1–2, pp. 81–89, 1978.



  • There are currently no refbacks.

Published by INSIGHT - Indonesian Society for Knowledge and Human Development