The exponentiated exponential distribution: a survey

Abstract

The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Renyi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.