Abstract

Although much progress has been made in the theory and application of bootstrap approximations

for max statistics in high dimensions, the literature has largely been restricted to cases involving

light-tailed data. To address this issue, we propose an approach to inference based on robust max

statistics, and we show that their distributions can be accurately approximated via bootstrapping

when the data are both high-dimensional and heavy-tailed. In particular, the data are assumed to

satisfy an extended version of the well-established L4-L2 moment equivalence condition, as well as

a weak variance decay condition. In this setting, we show that near-parametric rates of bootstrap

approximation can be achieved in the Kolmogorov metric, independently of the data dimension.

Moreover, this theoretical result is complemented by encouraging empirical results involving both

Euclidean and functional data.

Key words and phrases: high-dimensional statistics; robustness; bootstrap; simultaneous inference; median-of-means 1

Information

Preprint No.SS-2025-0036
Manuscript IDSS-2025-0036
Complete AuthorsMingshuo Liu, Miles Lopes
Corresponding AuthorsMiles Lopes
Emailsmelopes@ucdavis.edu

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Acknowledgments

We are grateful to Mengxin Yu for generously providing the code for the HL method.

Supplementary Materials

The supplementary materials contain the proofs of Theorem 1 and Proposition 1.

Supplementary materials are available for download.