Point Processes with a Generalized Order Statistic Property

Mixed Poisson processes are a well known class of point processes derived from (stationary) Poisson processes. In particular they cover cases where the intensity of a Poisson process is unknown but can be assumed to follow a known probability distribution. This situation is common e. g. in insurance mathematics where for instance the number of accident claims in which an individual is involved and which is evolving over some time can in principal be well described by a Poisson process with an individual, yet normally unknown intensity corresponding to the individual's accident proneness. Modelling this intensity as a random variable naturally leads to a mixed model. Usually, an insurance company will have a good estimate of the associated mixing distribution due to its large portfolio of policies.