A novel claim size distribution based on a Birnbaum–Saunders and gamma mixture capturing extreme values in insurance: estimation, regression, and applications

Data including significant losses are a pervasive issue in general insurance. The computation of premiums and reinsurance premiums, using deductibles, in situations of heavy right tail for the empirical distribution, is crucial. In this paper, we propose a mixture model obtained by compounding the Birnbaum–Saunders and gamma distributions to describe actuarial data related to financial losses. Closed-form credibility and limited expected value premiums are obtained. Moment estimators are utilized as starting values in the non-linear search procedure to derive the maximum-likelihood estimators and the asymptotic variance–covariance matrix for these estimators is determined. In comparison to other competing models commonly employed in the actuarial literature, the new mixture distribution provides a satisfactory fit to empirical data across the entire range of their distribution. The right tail of the empirical distribution is essential in the modeling and computation of reinsurance premiums. In addition, in this paper, to make advantage of all available data, we create a regression structure based on the compound distribution. Then, the response variable is explained as a function of a set of covariates using this structure.