  Wilk's theorem, which offers universal chi-squared approximations for
likelihood ratio tests, is widely used in many scientific hypothesis testing
problems. For modern datasets with increasing dimension, researchers have found
that the conventional Wilk's phenomenon of the likelihood ratio test statistic
often fails. Although new approximations have been proposed in high dimensional
settings, there still lacks a clear statistical guideline regarding how to
choose between the conventional and newly proposed approximations, especially
for moderate-dimensional data. To address this issue, we develop the necessary
and sufficient phase transition conditions for Wilk's phenomenon under popular
tests on multivariate mean and covariance structures. Moreover, we provide an
in-depth analysis of the accuracy of chi-squared approximations by deriving
their asymptotic biases. These results may provide helpful insights into the
use of chi-squared approximations in scientific practices.
