## Brazilian Journal of Probability and Statistics

### The exponentiated Kumaraswamy distribution and its log-transform

#### Abstract

The paper by Kumaraswamy (Journal of Hydrology 46 (1980) 79–88) introduced a probability distribution for double bounded random processes which has considerable attention in hydrology and related areas. Based on this distribution, we propose a generalization of the Kumaraswamy distribution refereed to as the exponentiated Kumaraswamy distribution. We derive the moments, moment generating function, mean deviations, Bonferroni and Lorentz curves, density of the order statistics and their moments. We also present a related distribution, so-called the log-exponentiated Kumaraswamy distribution, which extends the generalized exponential (Aust. N. Z. J. Stat. 41 (1999) 173–188) and double generalized exponential (J. Stat. Comput. Simul. 80 (2010) 159–172) distributions. We discuss maximum likelihood estimation of the model parameters. In applications to real data sets, we show that the log-exponentiated Kumaraswamy model can be used quite effectively in analyzing lifetime data.

#### Article information

Source
Braz. J. Probab. Stat., Volume 27, Number 1 (2013), 31-53.

Dates
First available in Project Euclid: 16 October 2012

https://projecteuclid.org/euclid.bjps/1350394628

Digital Object Identifier
doi:10.1214/11-BJPS149

Mathematical Reviews number (MathSciNet)
MR2991777

Zentralblatt MATH identifier
1319.62032

#### Citation

Lemonte, Artur J.; Barreto-Souza, Wagner; Cordeiro, Gauss M. The exponentiated Kumaraswamy distribution and its log-transform. Braz. J. Probab. Stat. 27 (2013), no. 1, 31--53. doi:10.1214/11-BJPS149. https://projecteuclid.org/euclid.bjps/1350394628

#### References

• Barreto-Souza, W. and Cribari-Neto, F. (2009). A generalization of the exponential-Poisson distribution. Statistics and Probability Letters 79, 2493–2500.
• Barreto-Souza, W., Santos, A. H. S. and Cordeiro, G. M. (2010). The beta generalized exponential distribution. Journal of Statistical Computation and Simulation 80, 159–172.
• Fletcher, S. C. and Ponnambalam, K. (1996). Estimation of reservoir yield and storage distribution using moments analysis. Journal of Hydrology 182, 259–275.
• Ganji, A., Ponnambalam, K. and Khalili, D. (2006). Grain yield reliability analysis with crop water demand uncertainty. Stochastic Environmental Research and Risk Assessment 20, 259–277.
• Garvan, F. (2002). The Maple Book. London: Chapman & Hall/CRC.
• Gilchrist, W. G. (2001). Statistical Modelling with Quantile Functions. Boca Raton, FL: Chapman & Hall/CRC.
• Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistic 41, 173–188.
• Hosking, J. R. M. (1990). $L$-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Ser. B 52, 105–124.
• Jones, M. C. (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology 6, 70–81.
• Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. New York: Springer-Verlag.
• Koutsoyiannis, D. and Xanthopoulos, T. (1989). On the parametric approach to unit hydrograph identification. Water Resources Management 3, 107–128.
• Kumaraswamy, P. (1980). Generalized probability density-function for double-bounded random-processes. Journal of Hydrology 46, 79–88.
• Lemonte, A. J. and Cordeiro, G. M. (2011). The exponentiated generalized inverse Gaussian distribution. Statistics and Probability Letters 81, 506–517.
• McDonald, J. B. (1984). Some generalized functions for the size distribution of income. Econometrica 52, 647–664.
• Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: Some properties and a flood data application. Communication in Statistics—Theory and Methods 25, 3059–3083.
• Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family. Technometrics 37, 436–445.
• Nadarajah, S. (2008). On the distribution of Kumaraswamy. Journal of Hydrology 348, 568–569.
• Nadarajah, S. and Kotz, S. (2006). The exponentiated type distributions. Acta Applicandae Mathematicae 92, 97–111.
• Nassar, M. M. and Eissa, F. H. (2003). On the exponentiated Weibull distribution. Communication in Statistics—Theory and Methods 32, 1317–1336.
• Ponnambalam, K., Seifi, A. and Vlach, J. (2001). Probabilistic design of systems with general distributions of parameters. International Journal of Circuit Theory and Applications 29, 527–536.
• R Development Core Team (2009). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
• Seifi, A., Ponnambalam, K. and Vlach, J. (2000). Maximization of manufacturing yield of systems with arbitrary distributions of component values. Annals of Operations Research 99, 373–383.
• Sigmon, K. and Davis, T. A. (2002). MATLAB Primer, 6th ed. New York: Chapman & Hall/CRC.
• Silva, R. B., Barreto-Souza, W. and Cordeiro, G. M. (2010). A new distribution with decreasing, increasing and upside-down bathtub failure rate. Computational Statistics and Data Analysis 54, 935–944.
• Song, K. S. (2001). Rényi information, loglikelihood and an intrinsic distribution measure. Journal of Statistical Planning and Inference 93, 51–69.
• Sundar, V. and Subbiah, K. (1989). Application of double bounded probability density-function for analysis of ocean waves. Ocean Engineering 16, 193–200.
• Wolfram, S. (2003). The Mathematica Book, 5th ed. New York: Cambridge Univ. Press.