Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size

TL;DR

This paper develops new sphericity tests for high-dimensional covariance matrices where the dimension greatly exceeds the sample size, establishing their asymptotic distributions and demonstrating their effectiveness through numerical experiments.

Contribution

It introduces a Quasi-LRT for high-dimensional sphericity testing and proves John's test maintains its distribution under various asymptotic regimes, extending applicability.

Findings

Quasi-LRT has a well-defined asymptotic distribution when p/n→∞.

John's test retains its distribution under all asymptotic regimes.

Numerical experiments confirm the theoretical properties and effectiveness.

Abstract

This paper focuses on the prominent sphericity test when the dimension $p$ is much lager than sample size $n$. The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when $p/n\rightarrow\infty, n\rightarrow\infty$ is well established in this paper. Meanwhile, John's test has been found to possess the powerful {\it dimension-proof} property, which keeps exactly the same limiting distribution under the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$, $n\rightarrow\infty$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.