Equilibrium compound distributions and stop-loss moments

A convolution representation is derived for the equilibrium or integrated tail distribution associated with a compound distribution. This result allows for the derivation of reliability properties of compound distributions, as well as an explicit analytic representation for the stop-loss premium, of interest in connection with insurance claims modelling. This result is extended to higher order equilibrium distributions, or equivalently to higher stop-loss moments. Special cases where the counting distribution is mixed Poisson or discrete phase-type are considered in some detail. An approach to handle more general counting distributions is also outlined. Keywords: Stop-loss premium, Integrated tail distribution, Compound geometric, Compound Poisson, Mixed Poisson, Beta distribution, Erlang distribution, Neyman class, Discrete phase-type, DS-NWUE, D-NWUE, D-NBUE, NWUE, NBUE