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 The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$.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 asymptotic distribution of the test statistic under the null when $p/n\to \infty ,n\to \infty$ 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. $p/n\to [0,\infty ]$, $n\to \infty$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.

References in corpus (1)

• On the sphericity test with large-dimensional observations