High-Dimensional Tests for Elliptical Models via Radial--Directional Dependence

TL;DR

This paper introduces new high-dimensional goodness-of-fit tests for elliptical models based on radial--directional independence, employing sum and max statistics for detecting departures.

Contribution

It develops novel tests that adapt to dense and sparse departures, with theoretical guarantees and practical diagnostics for elliptical models.

Findings

The tests control size well in simulations.

They effectively detect both dense and sparse alternatives.

Data analysis confirms interpretability and diagnostic utility.

Abstract

We develop high-dimensional goodness-of-fit tests for elliptical models by testing radial--directional independence after affine standardization. The method forms coordinatewise correlations between the log-radius and directional components, using a sum statistic for dense departures, a max statistic for sparse departures, and a Cauchy combination for adaptation. We derive oracle null limits, prove asymptotic independence of the sum and max components under both the null and a balanced local alternative, and establish validity of high-dimensional Hettmansperger--Randles plug-in standardization under explicit perturbation rates. Simulations and data analyses show stable size control, dense--sparse power complementarity, and interpretable coordinate-level diagnostics.