  Stein operators are differential operators which arise within the so-called
Stein's method for stochastic approximation. We propose a new mechanism for
constructing such operators for arbitrary (continuous or discrete) parametric
distributions with continuous dependence on the parameter. We provide explicit
general expressions for location, scale and skewness families. We also provide
a general expression for discrete distributions. For specific choices of target
distributions (including the Gaussian, Gamma and Poisson) we compare the
operators hereby obtained with those provided by the classical approaches from
the literature on Stein's method. We use properties of our operators to provide
upper and lower variance bounds (only lower bounds in the discrete case) on
functionals $h(X)$ of random variables $X$ following parametric distributions.
These bounds are expressed in terms of the first two moments of the derivatives
(or differences) of $h$. We provide general variance bounds for location, scale
and skewness families and apply our bounds to specific examples (namely the
Gaussian, exponential, Gamma and Poisson distributions). The results obtained
via our techniques are systematically competitive with, and sometimes improve
on, the best bounds available in the literature.
