A model for the number or amount of aggregate claim values on a portfolio of insurance business is analysed. The number of claims process is assumed to be a (possibly time transformed) mixed Poisson process, and the value of a claim is allowed to depend on the time of incurral as well as the end point of the observation period. The mixed Erlang assumption for claim amounts is seen to carry over to the aggregate claims fairly generally. Special cases of the model include the usual aggregate claims model with or without inflation, as well as a model for the incurred but not reported claims (IBNR), also with or without inflation. Connections between the inflation and IBNR models are established, and the notions of self-decomposability and discrete self-decomposability are seen to be relevant. Various examples are presented illustrating the ideas, and a numerical example is considered demonstrating how Panjer-type recursions may be employed to evaluate distributions of interest.
Compound distributionMixed PoissonMixed ErlangGamma distributionOrder statistic propertyInflationIncurred but not reportedSelf-decomposableDiscrete self-decomposableInfinite server MX/M/∞ queue