New Edgeworth-type expansions with finite sample guarantees

We establish higher-order expansions for a difference between probability distributions of sums of i.i.d. 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 obtained error terms depend on a sample size and a dimension explicitly. The new inequalities outperform accuracy of the normal approximation in existing Berry--Esseen inequalities under very general conditions. For symmetrically distributed random summands, the obtained results are optimal in terms of the ratio between the dimension and the sample size. Using the new higher-order inequalities, we study accuracy of the nonparametric bootstrap approximation and propose a bootstrap score test under possible model misspecification. The proposed results include also explicit error bounds for general elliptical confidence regions for an expected value of the random summands, and optimality of the Gaussian anti-concentration inequality over the set of all Euclidean balls.