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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] = [...]