Hu Zongliang, Tong Tiejun, Genton Marc G
College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China.
Department of Mathematics, Hong Kong Baptist University, Hong Kong.
Biometrics. 2019 Mar;75(1):256-267. doi: 10.1111/biom.12984. Epub 2019 Mar 6.
We propose a likelihood ratio test framework for testing normal mean vectors in high-dimensional data under two common scenarios: the one-sample test and the two-sample test with equal covariance matrices. We derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure. In comparison with the diagonal Hotelling's tests, our proposed test statistics display some interesting characteristics. In particular, they are a summation of the log-transformed squared t-statistics rather than a direct summation of those components. More importantly, to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses, we do not need the requirement that the covariance matrices follow a diagonal matrix structure. As a consequence, our proposed test methods are very flexible and readily applicable in practice. Simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods.
我们提出了一种似然比检验框架，用于在两种常见情况下检验高维数据中的正态均值向量：单样本检验和协方差矩阵相等的两样本检验。我们在协方差矩阵遵循对角矩阵结构的假设下推导检验统计量。与对角Hotelling检验相比，我们提出的检验统计量表现出一些有趣的特征。特别是，它们是对数变换后的平方t统计量的总和，而不是那些分量的直接总和。更重要的是，为了在原假设和局部备择假设下推导我们检验统计量的渐近正态性，我们不需要协方差矩阵遵循对角矩阵结构的要求。因此，我们提出的检验方法非常灵活，在实践中易于应用。还进行了模拟研究和实际数据分析，以证明我们似然比检验方法的优势。
