BOOTSTRAPPING MAX STATISTICS IN HIGH DIMENSIONS: NEAR-PARAMETRIC RATES UNDER WEAK VARIANCE DECAY AND APPLICATION TO FUNCTIONAL AND MULTINOMIAL DATA

• 成果类型: Article
• 署名作者: Lopes, Miles E.; Lin, Zhenhua; Mueller, Hans-Georg
• 署名单位: University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1844
• 发表日期: 2020
• 页码: 1214-1229
• 关键词: simultaneous confidence-intervals; Simultaneous Inference; spectral projectors; inequalities; approximations; bounds; tests

摘要

In recent years, bootstrap methods have drawn attention for their ability to approximate the laws of max statistics in high-dimensional problems. A leading example of such a statistic is the coordinatewise maximum of a sample average of n random vectors in R-p. Existing results for this statistic show that the bootstrap can work when n << p, and rates of approximation (in Kolmogorov distance) have been obtained with only logarithmic dependence in p. Nevertheless, one of the challenging aspects of this setting is that established rates tend to scale like n(-1/6) as a function of n. The main purpose of this paper is to demonstrate that improvement in rate is possible when extra model structure is available. Specifically, we show that if the coordinatewise variances of the observations exhibit decay, then a nearly n(-1/2) rate can be achieved, independent of p. Furthermore, a surprising aspect of this dimension-free rate is that it holds even when the decay is very weak. Lastly, we provide examples showing how these ideas can be applied to inference problems dealing with functional and multinomial data.