  In this paper, we propose a new test for testing the equality of two
population covariance matrices in the ultra-high dimensional setting that the
dimension is much larger than the sizes of both of the two samples. Our
proposed methodology relies on a data splitting procedure and a comparison of a
set of well selected eigenvalues of the sample covariance matrices on the split
data sets. Compared to the existing methods, our methodology is adaptive in the
sense that (i). it does not require specific assumption (e.g., comparable or
balancing, etc.) on the sizes of two samples; (ii). it does not need
quantitative or structural assumptions of the population covariance matrices;
(iii). it does not need the parametric distributions or the detailed knowledge
of the moments of the two populations. Theoretically, we establish the
asymptotic distributions of the statistics used in our method and conduct the
power analysis. We justify that our method is powerful under very weak
alternatives. We conduct extensive numerical simulations and show that our
method significantly outperforms the existing ones both in terms of size and
power. Analysis of two real data sets is also carried out to demonstrate the
usefulness and superior performance of our proposed methodology. An
$\texttt{R}$ package $\texttt{UHDtst}$ is developed for easy implementation of
our proposed methodology.
