New Edgeworth-type expansions with finite sample guarantees

TL;DR

This paper develops higher-order nonasymptotic distribution expansions for sums of i.i.d. random vectors, providing explicit error bounds that improve upon classical normal approximations, with applications to bootstrap methods and confidence regions.

Contribution

It introduces new Edgeworth-type expansions with finite sample guarantees, explicitly accounting for higher moments and improving accuracy over existing inequalities.

Findings

Bounds are uniform over Euclidean balls and half-spaces.

Error terms depend explicitly on sample size and dimension.

Results are optimal under symmetry assumptions.

Abstract

We establish higher-order nonasymptotic 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. Under some symmetry assumptions on the probability distribution of random summands, the obtained results are optimal in terms of the ratio between the dimension and the sample size. The new technique which we developed for establishing nonasymptotic higher-order expansions can be interesting by itself. 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 results of the paper also include explicit error bounds for general elliptic 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.