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 signal in the linear combination of the true mean vectors is weakly dense. 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.

翻译：本文通过随机积分方法研究了多个总体均值向量线性组合的假设检验问题。我们建立了检验统计量在原假设和备择假设下的渐近分布。此外，针对真实均值向量线性组合中非零信号呈弱密集分布的特殊情形，我们提供了检验统计量的理论解释。通过蒙特卡洛模拟将所提检验方法与现有高维检验进行对比，结果表明我们的检验在控制检验尺寸的同时展现出更优的检验功效。