A characterization of the distribution of a weighted sum of gamma variables through multiple hypergeometric functions

Abstract

Applying the theory on multiple hypergeometric functions, the distribution of a weighted convolution of gamma variables is characterized through explicit forms for the probability density function, the distribution function and the moments about the origin. The main results unify some previous contributions in the literature on finite convolution of gamma distributions. We deal with computational aspects that arise from the representations in terms of multiple hypergeometric functions, introducing a new integral representation for the fourth Lauricella function and its confluent form , suitable for numerical integration; some graphics of the probability density function and distribution function show that the proposed numerical approach supply good estimates for the special functions involved. We briefly outline two interesting applications of special function theory in statistics: the weighted convolutions of gamma matrices random variables and the weighted convolutions of gamma variables with random weights.

Keywords:

• weighted gamma convolution
• Lauricella functions
• confluent hypergeometric functions
• Dirichlet averages
• double Dirichlet averages
• multiple numerical integration

AMS Subject Classification :

• 33C65
• 33C70
• 60E10
• 62E17
• 65D20

Acknowledgements

I wish to thank Professor Gianfranco Lovison for many suggestions and careful reading of the paper.

Additional information

Notes on contributors

Francesca Di Salvo

*Email: [email protected]