High-Dimensional Two-Sample Test for Elliptical Symmetry Distribution

2026-05-04

AI Summary

This study addresses the failure of calibration and bias in diagonal standardization that plague high-dimensional two-sample location tests under elliptically symmetric distributions, particularly when strong covariance correlations and heavy-tailed data are present. To overcome these issues, the authors propose a novel spatial sign test based on pairwise coordinate-wise differences scaled by their marginal quantiles. The method employs a diagonal standardizer that requires no moment conditions on the radial component and accommodates arbitrary correlation structures, combined with diagonal deletion correction and a Rademacher wild bootstrap. Theoretically, the work establishes, for the first time under general dependence, a weighted chi-square null distribution and a high-dimensional stochastic expansion. Practically, it yields uniformly consistent null distribution estimation—subsuming the classical normal approximation as a special case when no dominant eigenvalues exist—and substantially enhances testing power in heavy-tailed, high-dimensional settings.

Abstract

We study the high-dimensional two-sample location problem under elliptical symmetry with arbitrary dependence in the scatter matrix. Existing spatial-sign procedures are attractive for heavy-tailed data, but their null calibration is tied to weakly dependent scatter matrices and their diagonal standardization does not, in general, recover the diagonal shape under strong dependence. We propose a new spatial-sign test based on coordinatewise pairwise-difference quantile scales. The new diagonal standardizer is location free, requires no positive moment condition on the radial variable, and estimates the diagonal of the elliptical shape up to a scalar specific to the sample, which disappears after spatial normalization. For the resulting full-sample statistic, we derive an explicit-rate stochastic expansion, establish a general weighted chi-square null distribution under arbitrary correlation structure, justify an empirical diagonal-deletion correction, and show that a Rademacher wild bootstrap consistently estimates the null law. The usual normal approximation appears only as a special case when no eigenvalue dominates.

Problem

• high-dimensional two-sample test
• elliptical symmetry
• spatial-sign test
• scatter matrix dependence
• diagonal standardization

Innovation

• spatial-sign test
• elliptical symmetry
• high-dimensional two-sample test
• diagonal standardization
• Rademacher wild bootstrap