  In this article, we study bounds on the uniform distance between the
cumulative distribution function of a standardized sum of independent centered
random variables with moments of order four and its first-order Edgeworth
expansion. Existing bounds are sharpened in two frameworks: when the variables
are independent but not identically distributed and in the case of independent
and identically distributed random variables. Improvements of these bounds are
derived if the third moment of the distribution is zero. We also provide
adapted versions of these bounds under additional regularity constraints on the
tail behavior of the characteristic function. We finally present an application
of our results to the lack of validity of one-sided tests based on the normal
approximation of the mean for a fixed sample size.
