Robust Max Statistics for High-Dimensional Inference

TL;DR

This paper introduces robust max statistics for high-dimensional inference with heavy-tailed data, demonstrating accurate bootstrap approximations under less restrictive conditions than traditional methods.

Contribution

It develops a new inference approach based on robust max statistics that work effectively with heavy-tailed data in high dimensions, extending existing bootstrap theory.

Findings

Bootstrap approximations achieve near-parametric rates in high dimensions.

The method performs well with both Euclidean and functional data.

Theoretical guarantees hold under extended moment and variance decay conditions.

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 $L^{4}$-$L^2$ 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.