  This article deals with Stein characterizations of probability distributions.
We provide a general framework for interpreting these in terms of the
parameters of the underlying distribution. In order to do so we introduce two
concepts (a class of functions and an operator) which generalize those which
were developed in the 70's by Charles Stein and Louis Chen for characterizing
the Gaussian and the Poisson distributions. Our methodology (i) allows for
writing many (if not all) known univariate Stein characterizations, (ii)
permits to identify clearly minimal conditions under which these results hold
and (iii) provides a straightforward tool for constructing new Stein
characterizations. Our parametric interpretation of Stein characterizations
also raises a number of questions which we outline at the end of the paper.
