Two-sided distributions with applications in insurance loss modeling

A framework of two-sided densities is presented for asymmetric continuous distributions consisting of two branches each with its own generating density. The framework supports the construction of distributions with positive support and a specified mode. Examples thereof shall be constructed using the beta and Burr Type XII distributions for their left and right branch densities. The examples are parameterized via left and right branch scale parameters, a mode parameter and two parameters determining the heaviness of its right tail. Keeping one of the tail parameters fixed, a procedure solving for their parameters is presented given a lower and upper quantile, a mode and a conditional-value-at-risk, popular in risk management of insurance losses. While valuable on its own right, that solution may be used as a starting point for a maximum likelihood routine. The estimation of the parameters is demonstrated using the classical insurance Danish fire loss data set and a French business loss interruption data set. Both data sets are publicly available. Developed models compare favorably with prior models fitted to the Danish fire loss data in the literature.