Adv. Appl. Prob. 42, 1013–1027 (2010) Printed in Northern Ireland © Applied Probability Trust 2010 ON ZERO-TRUNCATING AND MIXING POISSON DISTRIBUTIONS J. VALERO, ∗ M. PÉREZ-CASANY ∗∗ and J. GINEBRA, ∗∗∗ Technical University of Catalonia Abstract The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions. Keywords: Count variable; Poisson mixture; zero-truncated distribution; probability generating function; size-biased version; shifted version 2010 Mathematics Subject Classification: Primary 60E05 Secondary 60E10 1. Introduction Nonnegative integer (count) data are frequently overdispersed, which means that the vari- ability is larger than that expected under the Poisson model. This overdispersion is often a consequence of a lack of homogeneity of the sampled population, and can be modelled through a two-stage process in which the distribution of each count would be Poisson but with an expected value that changes randomly from count to count. By modelling the distribution of that expectation, one is naturally led to the use of Poisson mixture models. Poisson mixture models frequently used in practice include finite mixture models in which the mixing distribution has finite support, as well as infinite mixture models, such as the negative binomial model, the generalized inverse Gaussian–Poisson model, or the Tweedie–Poisson model. The literature dealing with these models and their applications goes back more than fifty years and is far too extensive to be covered adequately in this introduction. Apart from being overdispersed, count data often have a proportion of zeros that is larger than that expected from the initial model, which leads to the need for zero-modified models. There are also many instances where the value zero cannot be observed as a consequence of the method of ascertainment, which leads to the need for zero truncated count models. Received 5 October 2009; revision received 16 August 2010. Research partially supported by the Spanish Ministry of Education and Science and FEDER grants TIN2009-14560- C03-03, MTM2006-09920 and MTM2006-01477, and Generalitat de Catalunya through grant SGR-1187. ∗ Postal address: Department of Applied Mathematics III, Technical University of Catalonia, C. Esteve Terradas 8, 08860 Castelldefels, Spain. ∗∗ Postal address: Department of Applied Mathematics II and Data Management group, Technical University of Catalonia, C. Jordi Girona 1-3 08034 Barcelona, Spain. Email address: marta.perez@upc.edu ∗∗∗ Postal address: Department of Statistics and Operations Research, Technical University of Catalonia, Avda. Diagonal 647 08028 Barcelona, Spain. 1013 https://doi.org/10.1239/aap/1293113149 Published online by Cambridge University Press