Abstract
In this work the exact distribution of the linear combination of p independent logistic random variables is studied. It is shown that the exact distribution may be represented as a shifted infinite sum of independent random variables distributed as the difference of two independent Generalized Integer Gamma distributions. In addition, two near-exact approximations are developed for this distribution. Numerical studies are conducted to access the degree of precision and also the computational performance of these approximations. The developed methodology is used to derive near-exact approximations for the linear combination of independent generalized logistic random variables.
Highlights
The logistic distribution is an important distribution in statistics and is used in several areas of research
Problems related with the use of linear combinations of independent logistic random variables may arise naturally from the applications addressed in the above references when these are considered in the multivariate setting
The probability density and cumulative distribution functions for the linear combination of n independent logistic random variables were obtained in [18] in terms of the H-function, which is difficult to use in practice. [15] develops approximations for the distribution of the sum of random variables with a generalized logistic distribution for the independent and identically distributed case
Summary
The logistic distribution is an important distribution in statistics and is used in several areas of research. [15] develops approximations for the distribution of the sum of random variables with a generalized logistic distribution for the independent and identically distributed case. We will denote this fact by Y ∼ GLogistic(p, q) We present two results on the exact distribution of the linear combination of independent logistic random variables. We will show how it is possible to derive near-exact approximations using the result in Theorem 2
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