A note on the overdispersed Poisson family

In the Poisson model for loss reserving it is assumed that the incremental claims are independent and Poisson distributed with expectations being the product of two factors, depending on the occurrence year and the development year, respectively. It is well-known that maximum-likelihood estimation in the Poisson model yields the chain-ladder estimators of the expected ultimate aggregate claims. Recently, this result has been extended to overdispersed Poisson models. In the present paper, it is shown that every overdispersed Poisson model can be transformed into the Poisson model by rescaling all incremental claims by a common factor.