Two-sample test of high dimensional means under dependence

The paper considers in the high dimensional setting a canonical testing problem in multivariate analysis, namely testing the equality of two mean vectors. We introduce a new test statistic that is based on a linear transformation of the data by the precision matrix which incorporates the correlations between the variables. The limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives and enjoys certain optimality. A simulation study is carried out to examine the numerical performance of the test and to compare it with other tests given in the literature. The results show that the test proposed significantly outperforms those tests in a range of settings.