  Consider $k$ independent random samples from $p$-dimensional multivariate
normal distributions. We are interested in the limiting distribution of the
log-likelihood ratio test statistics for testing for the equality of $k$
covariance matrices. It is well known from classical multivariate statistics
that the limit is a chi-square distribution when $k$ and $p$ are fixed
integers. Jiang and Yang~\cite{JY13} and Jiang and Qi~\cite{JQ15} have obtained
the central limit theorem for the log-likelihood ratio test statistics when the
dimensionality $p$ goes to infinity with the sample sizes. In this paper, we
derive the central limit theorem when either $p$ or $k$ goes to infinity. We
also propose adjusted test statistics which can be well approximated by
chi-squared distributions regardless of values for $p$ and $k$. Furthermore, we
present numerical simulation results to evaluate the performance of our
adjusted test statistics and the log-likelihood ratio statistics based on
classical chi-square approximation and the normal approximation.
