Probabilistic characteristics of balance index of factors with generalized gamma distribution

The main probabilistic characteristics of balance index in Bayesian formulation, assuming that negative and positive factors have a priori generalized gamma distribution, are given. The formulation of this problem is equivalent to a study of generalized gamma laws scale mixture characteristics. Special attention is paid to the case in which the factors distributions have shape parameters of different signs. Moment characteristics and different presentation of density in terms of gamma-exponential function, H-function, Macdonald function, and generalized hypergeometric function are given. The analysis method is based on Mellin transform and its inverse transform. New properties of gamma-exponential function are given. The obtained results can be widely applied within the natural science models that use distributions with positive unlimited support to describe processes and phenomena.