发表机构
National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高维两样本协方差检验难题，提出保留零假设的变换结合学生化ℓ₂型统计量与高斯乘子自助法，经模拟和乳腺癌基因数据验证，可提升弱相关性下的检验水平准确性与功效。
AI 中文摘要
当两个高维协方差矩阵的许多元素仅存在微小差异时，检验二者的相等性颇具挑战性。样本协方差元素间的相关性也会影响那些聚合其差异的检验的有限样本水平和功效。我们提出了一种学生化的ℓ₂型统计量，该统计量对两个样本协方差矩阵对应元素之间经边际标准化的平方差异取平均。在构建该统计量之前，我们对两个样本应用相同的非奇异线性变换。在变换后的坐标系中，协方差相等的假设保持不变，而标准化差异及其估计量之间的相关性通常会发生变化。因此，该变换可纳入结构或科学信息，且不会降低维度。我们使用高斯乘子自助法来近似零分布，该方法采用逐坐标学生化，且无需构建或求向量化样本协方差元素的完整协方差矩阵。对于确定性变换，我们推导了非渐近高斯和自助法近似界，并确立了渐近水平有效性和功效一致性。我们还表明，在算子范数收敛条件下，将总体变换替换为同一样本估计量，会使统计量和自助法临界值在相对意义上渐近不变。模拟结果显示，合适的变换在弱相关性下可提升水平准确性和功效；对乳腺癌基因表达数据的分析也验证了该方法的有效性。
英文摘要
Testing equality of two high-dimensional covariance matrices is challenging when many entries differ only slightly. Dependence among sample covariance entries can also affect the finite-sample size and power of tests that aggregate their differences. We propose a studentized $\ell_2$-type statistic that averages squared, marginally standardized differences between corresponding entries of the two sample covariance matrices. Before constructing the statistic, we apply the same nonsingular linear transformation to both samples. In the transformed coordinates, the covariance-equality hypothesis is unchanged, whereas the standardized differences and correlations among their estimators generally change. The transformation can therefore incorporate structural or scientific information without reducing dimension. We approximate the null distribution using a Gaussian multiplier bootstrap that uses coordinatewise studentization and avoids forming or inverting the full covariance matrix of the vectorized sample covariance entries. For deterministic transformations, we derive nonasymptotic Gaussian and bootstrap approximation bounds and establish asymptotic size validity and power consistency. We also show that replacing a population transformation by a same-sample estimator leaves the statistic and bootstrap critical value asymptotically unchanged in relative terms under an operator-norm convergence condition. Simulations show that suitable transformations improve size accuracy and power under weak dependence. An analysis of breast cancer gene-expression data illustrates the method.
