Models for underreporting: a Bernoulli sampling approach for reported counts

Underreporting in register systems can be analyzed using a binomial approach, where both the size and the probability parameter have to be estimated. Parameter estimation fails when overdispersion is present. Extensions of the binomial model are derived by randomizing the parameters, i.e. considering mixed models. Among these models are the beta-binomial, which results from allowing for a random reporting probability; the negative-binomial, that is the marginal when the size parameter is randomized; and the beta-Poisson model, where both binomial parameters are considered random. The performance of these models is studied in simulation experiments and found to work well, safe for two exceptions. One is the Poisson limit, where the probability parameter approaches zero, and the second is the perfect system limit, where this probability approaches one. Finally the method is applied to real data from the Austrian crime register