Nonclassical Berry–Esseen inequalities and accuracy of the bootstrap

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We study accuracy of bootstrap procedures for estimation of quantiles of a smooth function of a sum of independent sub-Gaussian random vectors. We establish higher-order approximation bounds with error terms depending on a sample size and a dimension explicitly. These results lead to improvements of accuracy of a weighted bootstrap procedure for general log-likelihood ratio statistics. The key element of our proofs of the bootstrap accuracy is a multivariate higher-order Berry–Esseen inequality. We consider a problem of approximation of distributions of two sums of zero mean independent random vectors, such that summands with the same indices have equal moments up to at least the second order. The derived approximation bound is uniform on the sets of all Euclidean balls. The presented approach extends classical Berry–Esseen type inequalities to higher-order approximation bounds. The theoretical results are illustrated with numerical experiments.