Test for high-dimensional linear hypothesis of mean vectors via random integration

In this paper, we investigate hypothesis testing for the linear combination of mean vectors across multiple populations through the method of random integration. We have established the asymptotic distributions of the test statistics under both null and alternative hypotheses. Additionally, we provide a theoretical explanation for the special use of our test statistics in situations when the nonzero signals in the linear combination of the true mean vectors are weakly dense and nearly the same sign. Moreover, Monte Carlo simulations are presented to evaluate the suggested test against existing high-dimensional tests. The findings from these simulations reveal that our test not only aligns with the performance of other tests in terms of size but also exhibits superior power.