A mixed Poisson model with varying element sizes

The data of Bissell [A.F. Bissell, A negative binomial model with varying element sizes, Biometrika 59 (1972) 435–441] are counts of the number of flaws in rolls of textile fabric of different lengths. Given that the simple Poisson model is rejected due to overdispersion, we propose an analysis of the data by means of a general family of discrete laws, indexed by a parameter that distinguishes several well-known distributions. Our methodology provides some insight into the quality of the production and a relatively straightforward way to evaluate the probability distribution of the aggregate cost of the defects, avoiding the evaluation of convolutions.