This paper addresses the issue of testing sphericity and identity of high-dimensional population covariance matrix when the data dimension exceeds the sample size.The central limit theorem of the first four moments of eigenvalues of sample covariance matrix is derived using random matrix theory for generally distributed populations.Further,some desirable asymptotic properties of the proposed test statistics are provided under the null hypothesis as data dimension and sample size both tend to infinity.Simulations show that the proposed tests have a greater power than existing methods for the spiked covariance model.