An application of the Morgenstern family with standard two-sided power and gamma marginal distributions to the Bayes premium in the collective risk model

Abstract

The Bayes premium is a quantity of interest in the actuarial collective risk model, under which certain hypotheses are assumed. The usual assumption of independence among risk profiles is very convenient from a computational point of view but is not always realistic. Recently, several authors in the field of actuarial and operational risks have examined the incorporation of some dependence in their models. In this paper, we approach this topic by using and developing a Farlie-Gumbel-Morgenstern (FGM) family of prior distributions with specified marginals given by standard two-sided power and gamma distributions. An alternative Poisson-Lindley distribution is also used to model the count data as the number of claims. For the model considered, closed expressions of the main quantities of interest are obtained, which permit us to investigate the behavior of the Bayes premium under the dependence structure adopted (Farlie-Gumbel-Morgenstern) when the independence case is included.