  Testing the equality of the covariance matrices of two high-dimensional
samples is a fundamental inference problem in statistics. Several tests have
been proposed but they are either too liberal or too conservative when the
required assumptions are not satisfied which attests that they are not always
applicable in real data analysis. To overcome this difficulty, a
normal-reference test is proposed and studied in this paper. It is shown that
under some regularity conditions and the null hypothesis, the proposed test
statistic and a chi-square-type mixture have the same limiting distribution. It
is then justified to approximate the null distribution of the proposed test
statistic using that of the chi-square-type mixture. The distribution of the
chi-square-type mixture can be well approximated using a three-cumulant matched
chi-square-approximation with its approximation parameters consistently
estimated from the data. The asymptotic power of the proposed test under a
local alternative is also established. Simulation studies and a real data
example demonstrate that in terms of size control, the proposed test
outperforms the existing competitors substantially.
