Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications

In this paper we propose a new compound negative binomial distribution by mixing the p negative binomial parameter with an inverse Gaussian distribution and where we consider the reparameterization p=exp(-[lambda]). This new formulation provides a tractable model with attractive properties which make it suitable for application not only in the insurance setting but also in other fields where overdispersion is observed. Basic properties of the new distribution are studied. A recurrence for the probabilities of the new distribution and an integral equation for the probability density function of the compound version, when the claim severities are absolutely continuous, are derived. A multivariate version of the new distribution is proposed. For this multivariate version, we provide marginal distributions, the means vector, the covariance matrix and a simple formula for computing multivariate probabilities. Estimation methods are discussed. Finally, examples of application for both univariate and bivariate cases are given.