  Edgeworth expansion provides higher-order corrections to the normal
approximation for a probability distribution. The classical proof of Edgeworth
expansion is via characteristic functions. As a powerful method for
distributional approximations, Stein's method has also been used to prove
Edgeworth expansion results. However, these results assume that either the test
function is smooth (which excludes indicator functions of the half line) or
that the random variables are continuous (which excludes random variables
having only a continuous component). Thus, how to recover the classical
Edgeworth expansion result using Stein's method has remained an open problem.
In this paper, we develop Stein's method for two-term Edgeworth expansions in a
general case. Our approach involves repeated use of Stein equations, Stein
identities via Stein kernels, and a replacement argument.
