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

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 The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$.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 its asymptotic distribution of the test statistic under the null when $p/n\rightarrow\infty,n\rightarrow\infty$ is well established in this paper. We also re-examine the well-known John’s invariant test for sphericity in this ultra-dimensional setting. An amazing result from the paper states that John’s test statistic has exactly the same limiting distribution under the ultra-dimensional setting with under other high-dimensional settings known in the literature. Therefore, John’s test has been found to possess the powerful dimension-proof property, which keeps exactly the same limiting distribution under the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$, $n\rightarrow\infty$. Nevertheless, the asymptotic distribution of both test statistics under the alternative hypothesis with a general population covariance matrix is also derived and incorporates the null distributions as special cases. The power functions are presented and proven to converge to 1 as $p/n\rightarrow\infty,~n\rightarrow\infty,~n^{3}/p=O(1)$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented to illustrate the finite sample performance of the results.