  From the distributional characterizations that lie at the heart of Stein's
method we derive explicit formulae for the mass functions of discrete
probability laws that identify those distributions. These identities are
applied to develop tools for the solution of statistical problems. Our
characterizations, and hence the applications built on them, do not require any
knowledge about normalization constants of the probability laws. To demonstrate
that our statistical methods are sound, we provide comparative simulation
studies for the testing of fit to the Poisson distribution and for parameter
estimation of the negative binomial family when both parameters are unknown. We
also consider the problem of parameter estimation for discrete
exponential-polynomial models which generally are non-normalized.
