2017/05/26 by Hang Deng, Cun-Hui Zhang, Cun‐Hui Zhang +2 · 19 citations
Mathematics · #Advanced Statistical Methods and Models #Applied mathematics #Artificial intelligence #Bonferroni correction #Combinatorics #Computer science #Dimension (graph theory) #FOS: Computer and information sciences #Gaussian #Inference #Markov Chains and Monte Carlo Methods #Mathematics #Maxima #Methodology (stat.ME) #Sample size determination #Statistical Methods and Inference #Statistics #stat.ME
paper · pdf · doi:10.48550/arxiv.1705.09528
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2017/05/26 · openalex created_date 2017/06/05 · arxiv created 2020/01/10 · arxiv updated 2020/01/13 · openalex updated_date 2026/08/06
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 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 anti-concentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable.