On Left-truncating and Mixing Poisson Distributions

The distributions obtained by left-truncating at k a mixed Poisson distribution, kT-MP, those obtained by mixing previously left-truncated Poisson distributions, M-kTP; and those obtained by left-truncating at k a mixture of previously left-truncated Poisson distributions, kT-M-iTP, are characterized by means of their probability generating function. The kT-MP models are useful because they aproximate well the mechanism behind many count data generating processes, and because under the hypothesis that the mixing distribution has probability zero at zero, as in the continuous case, they allow one to estimate the first k + 1 probabilities of the untruncated mixed Poisson model. Consequences of the characterizations obtained are that every kT-MP distribution is a M-kTP distribution but not the other way around, and that the set of distributions kT-M-iTP is included in the set kT-M-(i+1)TP. Based on the characterizations obtained it follows that the factorial size-biased version of order k + 1 of a mixed Poisson random variable and, under a certain condition, its shifted version of order k + 1 are neither kT-MP nor M-kTP distributed. A transformation that applied to a mixed Poisson distribution always yields to a M-kTP distribution is defined.