Asymptotically optimal Berry–Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are offered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The method employed is a modification of the method of compositions along with Zolotarevʹs ideal metric. The approximation error in the CLT obtained vanishes at a rate O(n−k/2+1), provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k−1 ( k ⩾ 3 ) first moments with the standard normal distribution. Moreover, for the first time, these new uniform Berry–Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.