An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev's ideal zeta(3)-metric measuring the difference between expectations of sufficiently smooth functions, like vertical bar.vertical bar(3), of a sum of independent random variables X-1,...X-n with finite third-order moments and a sum of independent symmetric two-point random variablesY(i), isoscedastic to the X-i. In the homoscedastic case of equal variances, and in particular, in case of identically distributed X-1,...X-n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).