Computing the exact distribution of a linear combination of generalized logistic random variables and its applications

Abstract

We consider methods for numerical evaluation of the probability density function, cumulative distribution function and quantile function of a linear combination of n independent generalized logistic random variables. Here we derive the exact expressions for the probability density function and cumulative distribution function in terms of Fox's H-function. We also provide an algorithm for numerical calculation of Fox H- function and we compare its performance with a method based on the numerical inversion of the characteristic function. The numerical precision, efficiency and computing time of the proposed algorithm were considered. Applicability and comparison of the suggested approaches are illustrated by an application to financial returns.

Keywords:

• Fox H-function
• numerical comparison
• application in finance

Acknowledgments

The third author acknowledges that the work was partially supported by the Slovak Research and Development Agency, project APVV-18-0066, and by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences, projects VEGA 2/0081/19 and VEGA 2/0096/21.

Additional information

Funding

This work was supported by the Slovak Research and Development Agency [APVV-18-0066] and the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences, projects [VEGA 2/0081/19, VEGA 2/0096/21].