A supplement on CLT for LSS under a large dimensional generalized spiked covariance model

Central limit theorem (CLT) for linear spectral statistics (LSSs) is widely used in large scale statistical inference when the sample size n and dimension p both tend to infinity. However, there always exists discrepancy between the sample mean and sample variance, and asymptotic mean and asymptotic variance when the CLT is applied for an LSS under spiked models. A major portion of the discrepancy is from the spiked eigenvalues, which depends on the dimensions (p,n) and the magnitudes of the spikes. In order to eliminate such discrepancy, we propose in this paper a supplement to the CLT defined as Hp CLT for a class of LSSs of sample covariance matrices. Simulation results demonstrate the success of the Hp CLT and exhibit its superiority to the original ones in various situations.