For the general linear hypothesis testing problem for high-dimensional data, several interesting tests have been proposed in the literature. Most of them have imposed strong assumptions on the underlying covariance matrix so that their test statistics under the null hypothesis are asymptotically normally distributed. In practice, however, these strong assumptions may not be satisfied or hardly be checked so that these tests are often applied blindly in real data analysis. Their empirical sizes may then be much larger or smaller than the nominal size. For these tests, this is a size control problem which cannot be overcome via purely increasing the sample size to infinity. To overcome this difficulty, in this paper, a new normal-reference test using the centralized \(L^2\)-norm based test statistic with three cumulant matched chi-square approximation is proposed and studied. Some theoretical discussion and two simulation studies demonstrate that in terms of size control, the new normal-reference test performs very well regardless of if the high-dimensional data are nearly uncorrelated, moderately correlated, or highly correlated and it outperforms two existing competitors substantially. Two real high-dimensional data examples motivate and illustrate the new normal-reference test.

对于高维数据的一般线性假设检验问题，文献中提出了几个有趣的检验。他们中的大多数人对基础协方差矩阵强加了强有力的假设，以便他们在原假设下的检验统计量是渐近正态分布的。然而，在实践中，这些强有力的假设可能无法得到满足或难以检验，因此在实际数据分析中往往盲目地应用这些检验。它们的经验尺寸可能比标称尺寸大得多或小得多。对于这些测试，这是一个大小控制问题，不能通过单纯地将样本大小增加到无穷大来克服。为了克服这个困难，在本文中，使用集中式\(L^2\)提出并研究了基于三累积量匹配卡方近似的基于范数的检验统计量。一些理论讨论和两个模拟研究表明，在大小控制方面，无论高维数据是几乎不相关、中度相关还是高度相关，新的正态参考测试都表现得非常好，并且大大优于两个现有的竞争对手。两个真实的高维数据示例激发并说明了新的正常参考测试。