This article proposes three tests for proportionality hypotheses regrading high-dimensional covariance matrices. Compared with currently available tests in the literature that fail in situations involving a “large $p$ small $n$” or require knowledge of the underlying normal distributions, these tests are nonparametric, and do not require specifying any known distribution to derive asymptotic distributions under both the null hypothesis as well as an alternative hypothesis. The theoretical justification for the proposed tests is provided to ensure their validity, especially when the number of dimensions $p$ is larger than the sample size $n$. Numerical studies show that the proposed tests are adaptively powerful against dense as well as sparse alternatives for a wide range of dimensions and sample sizes. The tests were used to analyze a gene expression dataset to verify their effectiveness.

本文针对比例假设提出了三种检验，以重构高维协方差矩阵。与文献中现有的在“大型”情况下无法通过的测试相比$p$ 小 $ñ$”或需要了解基本的正态分布，这些检验是非参数的，并且不需要指定任何已知的分布即可在原假设和替代假设下得出渐近分布。提供了建议的测试的理论依据，以确保其有效性，尤其是在尺寸数量方面$p$ 大于样本量 $ñ$。数值研究表明，所提出的测试对于各种尺寸和样本量的密集和稀疏替代品具有自适应能力。这些测试用于分析基因表达数据集以验证其有效性。