Cauchy-Combined Hettmansperger-Randles Location Tests in High Dimensions

2026-10-03

AI Summary

This study addresses the failure of classical high-dimensional mean testing methods under strong dependence and heavy-tailed distributions, as well as their sensitivity to outliers. To overcome these limitations, this work proposes a robust regularized Hotelling testing framework. Methodologically, it integrates Hettmansperger–Randles spatial estimation with a shrinkage Tyler scatter matrix, leveraging ridge-resolvent random matrix theory to establish asymptotic normality without conventional sparsity assumptions. Furthermore, a Cauchy combination approach is employed to achieve adaptive testing across varying shrinkage levels. Simulation experiments and applications to gene expression data demonstrate that the proposed method substantially enhances robustness and accuracy in heavy-tailed scenarios while effectively preserving statistical power.

Abstract

High-dimensional mean testing is challenging under strong dependence and heavy tails, since classical Hotelling statistics are ill posed and regularized versions based on sample moments remain sensitive to outliers. This paper develops a robust regularized Hotelling framework for one-sample mean inference under elliptical distributions. The proposed HRST statistic combines a Hettmansperger-Randles spatial location estimator with the inverse of a trace-normalized shrinkage Tyler scatter matrix, and calibrates the resulting quadratic form through ridge-resolvent random-matrix theory rather than sparse covariance or precision-matrix assumptions. In the regime $p/n\to y\in(0,\infty)$, allowing a vanishing lower bulk edge and finite-rank diverging spikes, we establish feasible asymptotic normality for each shrinkage level, derive local-alternative distributions with explicit noncentrality parameters, and prove joint Gaussian limits over finite shrinkage grids. These results justify an adaptive Cauchy combination test across shrinkage levels. Simulations and a paired tumor-normal gene-expression study show that HRST maintains size and improves robustness under heavy-tailed distributions while retaining power under strong correlation.

Problem

• High-dimensional mean testing
• Heavy tails
• Strong dependence
• Robust Hotelling test
• Elliptical distributions

Innovation

• Robust regularized Hotelling test
• Tyler scatter matrix
• Random matrix theory
• Cauchy combination test
• High-dimensional mean testing