Correlation matrices are an essential tool for investigating the dependency structures of random vectors or comparing them. We introduce an approach for testing a variety of null hypotheses that can be formulated based upon the correlation matrix. Examples cover MANOVA-type hypothesis of equal correlation matrices as well as testing for special correlation structures such as, e.g., sphericity. Apart from existing fourth moments, our approach requires no other assumptions, allowing applications in various settings. To improve the small sample performance, a bootstrap technique is proposed and theoretically justified. Based on this, we also present a procedure to simultaneously test the hypotheses of equal correlation and equal covariance matrices. The performance of all new test statistics is compared with existing procedures through extensive simulations.

翻译：相关矩阵是研究随机向量依赖结构或比较其差异的重要工具。本文提出了一种检验基于相关矩阵构建的各类零假设的方法。实例涵盖MANOVA类型的等相关矩阵假设，以及检验特殊相关结构（如球度性）的假设。除现有四阶矩条件外，该方法无需其他假设，可适用于多种场景。为改进小样本性能，本文提出了一种bootstrap技术并提供了理论证明。在此基础上，我们还提出了一种同时检验等相关矩阵和等协方差矩阵假设的程序。通过大量模拟实验，我们将所有新检验统计量的性能与现有方法进行了比较。