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Title: Bootstrapping max statistics in high dimensions: Near-parametric rates and application to functional data analysis Authors:  Miles Lopes - UC Davis (United States) [presenting]
Zhenhua Lin - University of California, Davis (United States)
Hans-Georg Mueller - University of California Davis (United States)
Abstract: 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 coordinate-wise maximum of a sample average of $n$ random vectors in $R^p$. Existing results for this statistic show that bootstrap consistency can be achieved 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 is to demonstrate that improvement in rate is possible when extra model structure is available. Specifically, we show that if the coordinate-wise 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. As a numerical illustration, we show how these ideas can be used in the context of functional data analysis to construct simultaneous confidence intervals for the Fourier coefficients of a mean function.