Modifications of Wald's score tests on large dimensional covariance matrices structure

This paper considers testing the covariance matrices structure based on Wald's score test in large dimensional setting. The hypothesis $H_0: \Sigma =\Sigma_0$ for a given matrix $\Sigma_0$, which covers the identity hypothesis test and sphericity hypothesis test as the special cases, is reviewed by the generalized CLT (Central Limit Theorem) for the linear spectral statistics of large dimensional sample covariance matrices from Jiang(2015) . The proposed tests can be applicable for large dimensional non-Gaussian variables in a wider range. Furthermore, the simulation study is provided to compare the proposed tests with other large dimensional covariance matrix tests for evaluation of their performances. As seen from the simulation results, our proposed tests are feasible for large dimensional data without restriction of population distribution and provide the accurate and steady empirical sizes, which are almost around the nominal size.