  This paper considers the asymptotic power of likelihood ratio test (LRT) for
the identity test when the dimension p is large compared to the sample size n.
The asymptotic distribution of LRT under alternatives is given and an explicit
expression of the power is derived. A simulation study is carried out to
compare LRT with other tests. All these studies show that LRT is a powerful
test to detect eigenvalues around zero.
  Key words and phrases: Covariance matrix, High dimensional data, Identity
test, Likelihood ratio test, Power
