  In this work, we redefined two important statistics, the CLRT test (Bai
et.al., Ann. Stat. 37 (2009) 3822-3840) and the LW test (Ledoit and Wolf, Ann.
Stat. 30 (2002) 1081-1102) on identity tests for high dimensional data using
random matrix theories. Compared with existing CLRT and LW tests, the new tests
can accommodate data which has unknown means and non-Gaussian distributions.
Simulations demonstrate that the new tests have good properties in terms of
size and power. What is more, even for Gaussian data, our new tests perform
favorably in comparison to existing tests. Finally, we find the CLRT is more
sensitive to eigenvalues less than 1 while the LW test has more advantages in
relation to detecting eigenvalues larger than 1.
