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.

中文翻译：

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非经典 Berry-Esseen 不等式和 bootstrap 的准确性

我们研究了用于估计独立子高斯随机向量之和的平滑函数的分位数的引导程序的准确性。我们明确地根据样本大小和维度建立带有误差项的高阶近似边界。这些结果提高了一般对数似然比统计的加权自举程序的准确性。我们证明 bootstrap 准确性的关键要素是多元高阶 Berry-Esseen 不等式。我们考虑了两个零均值独立随机向量和的分布的近似问题，使得具有相同索引的被加数具有至少达到二阶的相等矩。导出的近似界限在所有欧几里得球的集合上是一致的。所提出的方法将经典的 Berry-Esseen 类型不等式扩展到高阶近似边界。理论结果用数值实验说明。