  This paper proposes a new test for covariance matrices structure based on the
correction to Rao's score test in large dimensional framework. By generalizing
the CLT for the linear spectral statistics of large dimensional sample
covariance matrices, the test can be applicable for large dimensional
non-Gaussian variables in a wider range without the restriction of the 4th
moment. Moreover, the amending Rao's score test is also powerful even for the
ultra high dimensionality as $p \gg n$, which breaks the inherent idea that the
corrected tests by RMT can be only used when $p<n$. Finally, we compare the
proposed test with other high dimensional covariance structure tests to
evaluate their performances through the simulation study.
