Mixtures of power series distributions: identifiability via uniqueness in problems of moments

We treat the identifiability problem for mixtures involving power series distributions. Applying an idea of Sapatinas (Ann Inst Stat Math 47:447–459, 1995) we prove and elaborate that a mixture distribution is identifiable if a certain Stieltjes problem of moments has a unique solution while a non-uniqueness leads to a non- identifiable mixture. We describe explicitly models of identifiable mixtures and models of non-identifiable mixtures. Illustrative examples and comments on related questions are also given.