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

2026-05-05

AI Summary

This study addresses the challenge of goodness-of-fit testing for high-dimensional elliptical models, where complex dependence between radial and directional components hinders conventional approaches. The authors propose an adaptive radial–directional dependence testing framework that first applies affine standardization to the data and then constructs test statistics based on correlations between log-radius and directional coordinates. By integrating sum-type, max-type, and Cauchy combination strategies, the method effectively detects dense, sparse, and mixed deviations from ellipticity, respectively. Theoretically, the work establishes, for the first time, the asymptotic independence of the sum and max statistics under both the null and local alternatives, and validates the efficacy of high-dimensional Hettmansperger–Randles plug-in standardization. Extensive simulations and real-data analyses demonstrate that the proposed procedure achieves accurate size control, complementary power across scenarios, and interpretable coordinate-level diagnostic capability.

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.

Problem

• high-dimensional
• elliptical models
• goodness-of-fit
• radial-directional independence
• affine standardization

Innovation

• elliptical models
• radial-directional independence
• high-dimensional goodness-of-fit test
• Cauchy combination
• affine standardization