A Damaged Generalised Poisson Model and its Application to Reported and Unreported Accident Counts

This paper investigates some models in which non-negative observations from a Poisson or generalised Poisson distribution are possibly damaged according to a binomial or quasi-binomial law. The latter case is appropriate when the observations are over-dispersed. Although the extent of the damage is not known, it is assumed that the event of whether or not damage occurred is discernible. The models are particularly suited for certain applications involving accident counts when evidence of certain accidents may be observed even though the accidents themselves may go unreported. Given the number of observed accidents and knowledge as to whether or not some additional accidents have gone unreported, these models may be used to make inferences concerning the actual number of unreported and total number of accidents in the current period, and the numbers of reported, unreported, and/or total accidents in a future period. The models are applied to a real data set giving reported and unreported patient accidents in a large hospital. Both maximum likelihood and Bayesian estimation methods are presented and discussed.