High-Dimensional Two-Sample Test for Elliptical Symmetry Distribution

TL;DR

This paper introduces a new high-dimensional two-sample test for elliptical distributions that is robust to dependence and heavy tails, with a novel diagonal standardizer and bootstrap calibration.

Contribution

It proposes a coordinatewise pairwise-difference quantile scale-based spatial-sign test that is location-free, does not require positive moments, and handles arbitrary dependence.

Findings

Derives an explicit stochastic expansion for the test statistic.

Establishes a weighted chi-square null distribution under arbitrary correlation.

Shows Rademacher wild bootstrap consistently estimates the null law.

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.