On the sphericity test with large-dimensional observations

In this paper, we propose corrections to LRT and John's test for sphericity in large-dimension. New formula for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are established. Using these formula, we derive the asymptotic distribution of the two proposed test statistics under the null without assuming parametric distribution for the population. Besides, simulations are conducted to analyze finite-sample performance of the corrected tests.