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New practically applicable estimates are constructed for the accuracy of approximation to the distributions of negative binomial random sums in the case where the shape parameter r > 0 of the negative binomial distribution satisfies the condition r [less than or equal to] [delta]/2, where 2 + [delta] is the maximum order of the moment of random summands, [delta] [member of] (0, 1]., 1. Introduction Let [X.sub.1], [X.sub.2], ... be independent identically distributed random variables with the distribution function F(x) = P([X.sub.1] < x), satisfying the moment conditions E[X.sub.1] = 0, [[sigma].sub.2] = [...]

We sharpen the upper bounds for the absolute constant in nonuniform convergence rate estimates in the central limit theorem for sums of independent identically distributed random variables possessing absolute moments of the order 2 + δ with some 0 < δ 1. In particular, it is demonstrated that under the existence of the third moment this constant does not exceed 18.2. Also it is shown that the absolute constant in the estimates under consideration can be replaced by a function C∗(∣x∣, δ) of the argument x of the difference of the prelimit and limit normal distribution functions for which a positive bounded nonincreasing majorant is found. Moreover, for δ = 1 this majorant is asymptotically exact (unimprovable) as x→∞and sharpens the estimates due to Nikulin [preprint, arXiv:1004.0552v1 [math.ST], 2010] for all x. For the first time a similar result is obtained for the case δ ∈ (0, 1). As a corollary, we obtain upper estimates for the Kolmogorov functions which are the analogues of the exact and the asymptotically exact constants in the (uniform) Berry–Esseen inequality. [ABSTRACT FROM AUTHOR]