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The exponentiated exponential distribution: a survey

机译：指数指数分布：调查

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The exponentiated exponential distribution, a most attractive generaliza tion 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 sur vey 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 or der 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 re search.
机译：Gupta和Kundu（Aust。N. Z. J. Stat。41：173-188，1999）引入的指数式指数分布是指数分布的最吸引人的概括。但是，这种分布的许多数学性质似乎是未知的，或者以更简单/一般的形式未知。在本文中，我们提供了数学特性的综合调查。我们导出矩生成函数，特征函数，累积量生成函数，第n个矩，前四个矩，方差，偏度，峰度，第n个条件矩，前四个累积量，关于均值的均值偏差，关于中位数，Bonferroni曲线，Lorenz曲线，Bonferroni浓度指数，Gini浓度指数，Renyi熵，Shannon熵，累积残差熵，Song测度，阶次统计量矩，L矩，极值或der统计量的渐近分布，可靠性，分布指数指数随机变量的总和，指数指数随机变量乘积的分布以及指数指数随机变量的比率分布。我们还讨论了采用最大似然方法进行的估计，包括审查的情况，并为Fisher信息矩阵提供了比Gupta和Kundu给出的表达式更简单的表达式。期望本文可以作为指数指数分布的参考资料，并鼓励进一步的研究。

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