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Abstract:
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In recent years, bootstrap methods have attracted considerable attention for their ability to approximate the laws of "max statistics" in high-dimensional problems. A canonical example of such a statistic is the coordinate-wise maximum of a sample average of high-dimensional random vectors. From a theoretical standpoint, existing results for this statistic show that bootstrap consistency can often be achieved when the sample size n and data dimension p satisfy n< < p. Despite this favorable dependence on p, a drawback of such results is that the rates of consistency (in Kolmogorov distance) often have a slow dependence on n, such as n^{-1/6} or n^{-1/8}. In this paper, we show that if the coordinate-wise variances of the observations exhibit decay, then a nearly parametric rate can be achieved, independent of the dimension. As an illustration, we show how it is possible to take advantage of variance decay when max statistics are used to construct simultaneous confidence intervals in the context of functional data.
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