Identity tests for high dimensional data using RMT

In this work, we redefined two important statistics, the CLRT test [Z. Bai, D. Jiang, J. Yao, S. Zheng, Corrections to LRT on large-dimensional covariance matrix by RMT, The Annals of Statistics 37 (6B) (2009) 3822–3840] and the LW test [O. Ledoit, M. Wolf, Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size, The Annals of Statistics (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.