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much lager than sample size n. The classical likelihood ratio test(LRT) is no
longer applicable when p≫n. Therefore a Quasi-LRT is proposed and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