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

Likelihood-based tests on linear hypotheses of large dimensional mean vectors with unequal covariance matrices