On the Distribution of the Random Sum and Linear Combination of Independent Exponentiated Exponential Random Variables

The exponentiated exponential distribution has received great attention from many statisticians due to its popularity, many applications, and the fact that it is an efficient alternative to many famous distributions such as Weibull and gamma distributions. Many statisticians have studied the mathematical properties of this distribution and estimated its parameters under different censoring schemes. However, it seems that the distribution of the random sum, the distribution of the linear combination, and the value of the reliability index, R=P(X2<X1), in the case of unequal scale parameters, were not known for this distribution. Therefore, in this article, we present the saddlepoint approximation to the distribution of the random sum, the distribution of linear combination, and the value of the reliability index R=P(X2<X1) for exponentiated exponential variates. These saddlepoint approximations are computationally appealing, and numerical studies confirm their accuracy. In addition to the accuracy provided by the saddlepoint approximation method, it saves time compared to the simulation method, which requires a lot of time. Therefore, the saddlepoint approximation method provides an outstanding balance between precision and computational efficiency.