High-Dimensional Two-Sample Test for Elliptical Symmetry Distribution
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.
Cite
@article{arxiv.2605.03265,
title = {High-Dimensional Two-Sample Test for Elliptical Symmetry Distribution},
author = {Long Feng and Hongfei Wang},
journal= {arXiv preprint arXiv:2605.03265},
year = {2026}
}