Gamma-Generalized Inverse Gaussian Class of Distributions with Applications

Abstract

In this article, a new family of probability distributions with domain in ℝ⁺ is introduced. This class can be considered as a natural extension of the exponential-inverse Gaussian distribution in Bhattacharya and Kumar (1986) and Frangos and Karlis (2004). This new family is obtained through the mixture of gamma distribution with generalized inverse Gaussian distribution. We also show some important features such as expressions of probability density function, moments, etc. Special attention is paid to the mixture with the inverse Gaussian distribution, as a particular case of the generalized inverse Gaussian distribution. From the exponential-inverse Gaussian distribution two one-parameter family of distributions are obtained to derive risk measures and credibility expressions. The versatility of this family has been proven in numerical examples.

Keywords:

• Claim amount
• Covariable
• Credibility
• Fit
• Gamma distribution
• Generalized inverse Gaussian distribution
• Posterior distribution

Mathematics Subject Classification:

• 62E10
• 62F10

Acknowledgments

We are indebted to the Editor and referees for helpful comments which improved an earlier version of the work. The authors thank the Ministerio de Ciencia y Tecnología for partial support of this work (EGD, projects SEJ2006-12685, MEC, Spain and ECO2009-14152, MICINN, Spain; JMS, project ECO2010-15455).