New Edgeworth-type expansions with finite sample guarantees
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We establish Edgeworth-type expansions for a difference between probability distributions of sums of independent random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean balls and the set of all half-spaces. These results allow to account for an impact of higher-order moments or cumulants of the considered distributions; the derived error terms depend on a sample size and a dimension explicitly. We compare these results with known Berry-Esseen inequalities, and show how the new bounds can outperform accuracy of the normal approximation. We apply the new bounds to the linear regression model and the smooth function model, and examine dependence of accuracy of the normal approximation in these models on higher-order moments, a dimension, and a sample size.
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