Cauchy-Combined Hettmansperger-Randles Location Tests in High Dimensions
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→ y∈(0,∞), 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.
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BibTeX
@misc{zhao2026cauchycombined,
title = {{Cauchy-Combined Hettmansperger-Randles Location Tests in High Dimensions}},
author = {Ping Zhao and Long Feng and Xiaoyi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04251},
eprint = {2610.04251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04251},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cauchy-Combined Hettmansperger-Randles Location Tests in High Dimensions
AU - Ping Zhao
AU - Long Feng
AU - Xiaoyi Wang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04251
UR - https://arxiv.org/abs/2610.04251
AB - 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→ y∈(0,∞), 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.
ER -