BEYOND GAUSSIAN APPROXIMATION: BOOTSTRAP FOR MAXIMA OF SUMS OF INDEPENDENT RANDOM VECTORS
BEYOND GAUSSIAN APPROXIMATION: BOOTSTRAP FOR MAXIMA OF SUMS OF INDEPENDENT RANDOM VECTORS
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DOI：
10.1214/20-aos1946
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发表时间：
2020-12-01
影响因子：
4.5
通讯作者：
Zhang, Cun-Hui
中科院分区：
文献类型：
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作者：
Deng, Hang;Zhang, Cun-Hui
Bonferroni平差或并界常用于研究高维问题中统计方法的速率最优性。然而，在实践中，Bonferroni调整过于保守。极值理论已被证明在许多情况下提供了更准确的多重性调整，但仅限于临时基础上。最近，当n>；>；(Logp)(7)(其中p是推理问题的多重性，n是样本量)时，在一些一般情况下，高斯近似被用来证明大规模同时推理中的Bootstrap调整。这一理论的主旨是高维独立随机向量和的最大值的高斯近似的有效性。本文将Kolmogorov-Smirnov距离下的经验自举和乘子/野生自举的一致性的样本量要求降低到n>；>；(Logp)(5)，可能是在高斯近似不可用的情况下。当现有的与高斯近似交织在一起的比较和反集中定理不再适用或足够强以产生期望的结果时，新的比较和反集中定理被发展起来。
The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on an ad hoc basis. Recently, Gaussian approximation has been used to justify bootstrap adjustments in large scale simultaneous inference in some general settings when n >> (log p)(7), where p is the multiplicity of the inference problem and n is the sample size. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high dimension. In this paper, we reduce the sample size requirement to n >> (log p)(5) for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov-Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anticoncentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable or strong enough to produce desired results.