  We propose a likelihood ratio test framework for testing normal mean vectors
in high-dimensional data under two common scenarios: the one-sample test and
the two-sample test with equal covariance matrices. We derive the test
statistics under the assumption that the covariance matrices follow a diagonal
matrix structure. In comparison with the diagonal Hotelling's tests, our
proposed test statistics display some interesting characteristics. In
particular, they are a summation of the log-transformed squared $t$-statistics
rather than a direct summation of those components. More importantly, to derive
the asymptotic normality of our test statistics under the null and local
alternative hypotheses, we do not need the requirement that the covariance
matrices follow a diagonal matrix structure. As a consequence, our proposed
test methods are very flexible and readily applicable in practice. Simulation
studies and a real data analysis are also carried out to demonstrate the
advantages of our likelihood ratio test methods.
