Tests for Large Dimensional Covariance Structure Based on Rao's Score Test

TL;DR

This paper introduces a novel covariance matrix test based on Rao's score, applicable to large, non-Gaussian, and ultra-high dimensional data, overcoming previous limitations related to moment restrictions and dimensionality.

Contribution

It generalizes the CLT for spectral statistics, enabling Rao's score test to work effectively in ultra-high dimensional settings without moment restrictions.

Findings

The proposed test performs well in simulations compared to existing methods.

It remains powerful even when the number of variables greatly exceeds the sample size.

The method extends applicability to non-Gaussian and ultra-high dimensional data.

Abstract

This paper proposes a new test for covariance matrices structure based on the correction to Rao's score test in large dimensional framework. By generalizing the CLT for the linear spectral statistics of large dimensional sample covariance matrices, the test can be applicable for large dimensional non-Gaussian variables in a wider range without the restriction of the 4th moment. Moreover, the amending Rao's score test is also powerful even for the ultra high dimensionality as $p \gg n$, which breaks the inherent idea that the corrected tests by RMT can be only used when $p<n$. Finally, we compare the proposed test with other high dimensional covariance structure tests to evaluate their performances through the simulation study.