Elliptical Regularized Hotelling Testing for High Dimensional Data

2026-06-24

AI Summary

This study addresses the challenge of one-sample location testing in high-dimensional heavy-tailed data with strong cross-sectional dependence. The authors propose the Elliptically Regularized Hotelling Test with Cauchy Combination (ERHT–CC), which constructs a regularized test statistic based on the spatial median and sign covariance matrix, and adaptively aggregates p-values across multiple ridge parameters using the Cauchy combination method—eliminating the need to estimate cross-ridge correlations or tune hyperparameters. Theoretically, this work establishes, for the first time under elliptically symmetric heavy-tailed distributions, a regularized Hotelling test with an explicit local power function and proves its asymptotic normality under the null hypothesis. Numerical experiments demonstrate that ERHT–CC substantially outperforms existing methods in scenarios involving heavy tails and strong dependence, exhibiting excellent finite-sample performance.

Abstract

We consider one-sample testing of a high-dimensional location parameter under elliptically symmetric distributions with heavy tails and pervasive cross-sectional dependence. We propose an elliptical regularized Hotelling test with Cauchy combination (ERHT--CC), based on the sample spatial median and the spatial-sign covariance matrix centered at that median. We derive its null asymptotic normality, consistent estimators of the centering and variance, and an explicit local power function. Since the power-optimal ridge parameter depends on the unknown alternative, we aggregate fixed-ridge $p$-values over a deterministic grid using the Cauchy rule. We establish a finite-grid joint Gaussian limit, justify the analytic combined $p$-value without estimating cross-ridge correlations, and characterize its local power. Simulation studies and an empirical analysis demonstrate the favorable finite-sample performance of ERHT--CC under heavy tails and pervasive dependence.

Problem

• high-dimensional testing
• elliptical distributions
• heavy tails
• cross-sectional dependence
• location parameter

Innovation

• elliptical regularization
• Cauchy combination
• spatial median
• high-dimensional testing
• heavy-tailed distributions