On linear combination of generalized logistic random variables with an application to financial returns

Abstract

We derive two expressions of the cumulative distribution function for the linear combination Z=αX+βY in case when X and Y are independent generalized logistic random variables. While the first expression is given in terms of infinite sums, the second expression is exact and it is given via the well known Fox H function. The exact cumulative distribution function of Z is derived by using Mellin and inverse Mellin transforms. We also consider two dependent logistic random variables case via Gumbel’s Type I bivariate logistic distribution and derive probability density function of the linear combination. The derived density function is found in elementary mathematical functions. In order to provide percentage points, we develop the numerical routine for calculation of the values of Fox H function. We study the application of the considered linear combination in the field of financial returns.