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April 2020 Bootstrapping max statistics in high dimensions: Near-parametric rates under weak variance decay and application to functional and multinomial data
Miles E. Lopes, Zhenhua Lin, Hans-Georg Müller
Ann. Statist. 48(2): 1214-1229 (April 2020). DOI: 10.1214/19-AOS1844

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 coordinatewise maximum of a sample average of $n$ random vectors in $\mathbb{R}^{p}$. Existing results for this statistic show that the bootstrap can work when $n\ll p$, 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.

Citation

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Miles E. Lopes. Zhenhua Lin. Hans-Georg Müller. "Bootstrapping max statistics in high dimensions: Near-parametric rates under weak variance decay and application to functional and multinomial data." Ann. Statist. 48 (2) 1214 - 1229, April 2020. https://doi.org/10.1214/19-AOS1844

Information

Received: 1 July 2018; Revised: 1 January 2019; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241587
MathSciNet: MR4102694
Digital Object Identifier: 10.1214/19-AOS1844

Subjects:
Primary: 62G09, 62G15
Secondary: 62G05, 62G20

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • April 2020
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