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This paper focuses on the prominent sphericity test when the dimension $p$ ismuch lager than sample size $n$. The classical likelihood ratio test(LRT) is nolonger applicable when $p\gg n$. Therefore a Quasi-LRT is proposed andasymptotic 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 {\itdimension-proof} property, which keeps exactly the same limiting distributionunder the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$,$n\rightarrow\infty$. All asymptotic results are derived for general populationwith finite fourth order moment. Numerical experiments are implemented forcomparison.

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