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

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sphericity test when the dimension p is much lager than sample size n. The classical likelihood ratio test(LRT) is no longer applicable when pgg n. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when p/nrightarrow∞, nrightarrow∞ is well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/nrightarrow[0,∞], nrightarrow∞. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.