Testing the equality between two high-dimensional covariance matrices is challenging. As the efficient way to measure evidential discrepancy from observed data, the likelihood ratio test is expected to be powerful when the null hypothesis is violated. However, when the data dimensionality becomes large and may substantially exceed the sample size, likelihood ratio based approaches are encountering both practical and theoretical difficulties. To solve the problem, we propose in this study to first randomly project the original high-dimensional data to some lower-dimensional space, and then to apply the corrected likelihood ratio tests developed with the random matrix theory. We show that our test is consistent under the null hypothesis. Through evaluating the power function which is a challenging objective in this context, we show evidence that our test based on random projection matrix with reasonable column size is more powerful when the two covariance matrices are unequal but component-wise discrepancy could be small—a weak and dense signal setting. Numerical studies with simulations and a real data analysis confirm the merits of our test.

中文翻译：

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用随机投影和校正似然比测试高维协方差矩阵

测试两个高维协方差矩阵之间的相等性具有挑战性。作为从观察数据中衡量证据差异的有效方法，当违反原假设时，似然比检验有望发挥作用。然而，当数据维度变得很大并且可能大大超过样本量时，基于似然比的方法会遇到实际和理论困难。为了解决这个问题，我们在本研究中建议首先将原始高维数据随机投影到一些低维空间，然后应用随机矩阵理论开发的校正似然比检验。我们证明我们的检验在原假设下是一致的。通过评估在这种情况下具有挑战性的目标的幂函数，我们证明了我们基于具有合理列大小的随机投影矩阵的测试在两个协方差矩阵不相等但组件方面的差异可能很小（弱且密集的信号设置）时更强大。带有模拟和真实数据分析的数值研究证实了我们测试的优点。