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

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