  In this paper, we propose corrections to the likelihood ratio test and John's
test for sphericity in large-dimensions. New formulas for the limiting
parameters in the CLT for linear spectral statistics of sample covariance
matrices with general fourth moments are first established. Using these
formulas, we derive the asymptotic distribution of the two proposed test
statistics under the null. These asymptotics are valid for general population,
i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive
Monte-Carlo experiments are conducted to assess the quality of these tests with
a comparison to several existing methods from the literature. Moreover, we also
obtain their asymptotic power functions under the alternative of a spiked
population model as a specific alternative.
