Two-sample tests for principal eigenvalues and eigenvectors in high-dimensional elliptical factor models

2026-09-29

AI Summary

This study addresses the failure of covariance inference and the instability of principal eigenstructures in high-dimensional factor models under heavy-tailed distributions. We propose a two-sample testing procedure for principal eigenvalues and eigenvectors that requires no higher-order moment assumptions. Built upon Tyler’s M-estimator, the method integrates orthogonalization, cross-fitting, and spectral asymptotic analysis to establish a robust limit theory applicable across diverse radial distributions. Our theoretical results demonstrate that the proposed approach significantly outperforms conventional covariance-based benchmarks when handling heavy-tailed data, enabling precise discrimination between changes in magnitude and direction. The practical effectiveness of this methodology is further validated through an empirical application to S&P 500 returns.

Abstract

Changes in principal eigenvalues and eigendirections provide complementary diagnostics of structural instability in factor models, but covariance-based inference can be unreliable under heavy tails. We develop two-sample tests for equality of these features of trace-normalized shape matrices under high-dimensional elliptical factor models. Our approach combines Tyler's identity with orthogonalization and cross-fitting to accommodate general location and precision pilots. We derive a spectral limit theory that yields asymptotically valid calibration without upper-tail moment assumptions on the radial variables, allowing the two populations to have different radial distributions and factor ranks. Simulations demonstrate good size control and gains in size-adjusted power over covariance-based benchmarks under heavy-tailed elliptical distributions. An application to S\&P~500 stock returns illustrates how the tests distinguish changes in relative component strength from changes in component orientation.

Problem

• high-dimensional elliptical factor models
• two-sample tests
• principal eigenvalues
• eigenvectors
• heavy-tailed distributions

Innovation

• Two-sample tests
• High-dimensional elliptical factor models
• Shape matrix
• Cross-fitting
• Spectral limit theory