In this paper, we propose a novel approach to test the equality of high-dimensional mean vectors of several populations via the weighted \(L_2\) -norm. We establish the asymptotic normality of the test statistics under the null hypothesis. We also explain theoretically why our test statistics can be highly useful when the nonzero signal in mean vectors is weakly dense with almost the same sign. Furthermore, we compare the proposed test with existing tests using simulation results, demonstrating that the weighted \(L_2\) -norm-based test statistic exhibits favorable properties in terms of both size and power.
