## Brazilian Journal of Probability and Statistics

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

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

Artur J. Lemonte, Wagner Barreto-Souza, and Gauss M. Cordeiro

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/11-BJPS149

**Mathematical Reviews number (MathSciNet)**

MR2991777

**Zentralblatt MATH identifier**

1319.62032

**Keywords**

Beta distribution Kumaraswamy distribution maximum likelihood estimation mean deviation order statistic

#### 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