  This paper considers testing linear hypotheses of a set of mean vectors with
unequal covariance matrices in large dimensional setting. The problem of
testing the hypothesis $H_0 : \sum_{i=1}^q \beta_i \bmu_i =\bmu_0 $ for a given
vector $\bmu_0$ is studied from the view of likelihood, which makes the
proposed tests more powerful. We use the CLT for linear spectral statistics of
a large dimensional $F$-matrix in Zheng(2012) [21] to establish the new test
statistics in large dimensional framework, so that the proposed tests can be
applicable for large dimensional non-Gaussian variables in a wider range.
Furthermore, our new tests provide more optimal empirical powers due to the
likelihood-based statistics, meanwhile their empirical sizes are closer to the
significant level. Finally, the simulation study is provided to compare the
proposed tests with other high dimensional mean vectors tests for evaluation of
their performances.
