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

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.