论文标题
Fisher的高维协方差矩阵概率测试
Fisher's combined probability test for high-dimensional covariance matrices
论文作者
论文摘要
测试大量协方差矩阵在具有高维数据的统计分析中至关重要。在过去的十年中，在文献中研究了三种类型的测试统计数据：二次形式统计，最大形式统计及其加权组合。众所周知，二次形式的统计数据将遭受稀疏替代方案的低功率，而最大形式的统计数据将遭受低功率，而较低的替代方案将遭受较低的功率。引入了加权组合方法，以提高二次形式统计的功率或当重量适当选择权重时的最大形式统计量。在本文中，我们提供了一种新的观点，以利用二次形式统计数据的全部潜力和最大形式统计数据来测试高维协方差矩阵。我们根据Fisher的方法提出了比例不变的功率增强测试，以结合二次形式统计的P值和最大形式统计。在仔细研究了二次形式统计和最大形式统计的渐近关节分布之后，我们证明所提出的组合方法保留了正确的渐近尺寸，并增强了对更一般替代方案的功率。此外，我们证明了模拟研究和实际应用中的有限样本性能。
Testing large covariance matrices is of fundamental importance in statistical analysis with high-dimensional data. In the past decade, three types of test statistics have been studied in the literature: quadratic form statistics, maximum form statistics, and their weighted combination. It is known that quadratic form statistics would suffer from low power against sparse alternatives and maximum form statistics would suffer from low power against dense alternatives. The weighted combination methods were introduced to enhance the power of quadratic form statistics or maximum form statistics when the weights are appropriately chosen. In this paper, we provide a new perspective to exploit the full potential of quadratic form statistics and maximum form statistics for testing high-dimensional covariance matrices. We propose a scale-invariant power enhancement test based on Fisher's method to combine the p-values of quadratic form statistics and maximum form statistics. After carefully studying the asymptotic joint distribution of quadratic form statistics and maximum form statistics, we prove that the proposed combination method retains the correct asymptotic size and boosts the power against more general alternatives. Moreover, we demonstrate the finite-sample performance in simulation studies and a real application.
