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 in weakly dense cases when the nonzero signal in mean vectors is present. 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.

翻译：本文提出了一种基于加权$L_2$范数检验多个总体高维均值向量相等性的新方法。我们在原假设下证明了检验统计量的渐近正态性,并从理论上解释了为何当均值向量中存在非零信号时,该检验统计量在弱稠密情况下具有高度实用性。此外,通过模拟结果将所提检验与现有方法进行对比,表明基于加权$L_2$范数的检验统计量在检验水平与检验功效方面均表现出优良特性。