AI 中文总结
该研究将2019年提出的关联矩阵商仿射黎曼几何与1970年Jennrich的关联矩阵相等性渐近检验关联，推导得出相关收敛结果并验证了p=2时的特例。
AI 中文摘要
商仿射度量为满秩关联矩阵赋予了内在的黎曼几何，但其测地距离无闭合形式，且目前尚无已知的解析渐近零分布。我们将这一2019年提出的几何与Jennrich于1970年提出的关联矩阵相等性渐近检验建立关联。Jennrich统计量所基于的二次型恰好是商仿射度量张量的一半。该等式的出现是因为从高斯费希尔信息中剔除边缘标准差的操作，与剔除对角缩放的商投影操作完全相同。因此，Jennrich统计量可直接评估局部商仿射二次型。此外，对于具有共同总体关联矩阵的两个独立高斯样本，经有效样本量缩放后的平方测地距离依分布收敛于$4\chi^2_d$，其中$d = p(p-1)/2$。当$p=2$时，该结果简化为两样本Fisher z检验。
英文摘要
The quotient-affine metric gives an intrinsic Riemannian geometry to full-rank correlation matrices, but its geodesic distance has no closed form and we are not aware of an analytic asymptotic null distribution for it. We connect this geometry, introduced in 2019, with Jennrich's 1970 asymptotic test for equality of correlation matrices. The quadratic form underlying Jennrich's statistic is exactly one half of the quotient-affine metric tensor. The identity arises because eliminating marginal standard deviations from Gaussian Fisher information performs the same projection as quotienting out diagonal rescalings. Jennrich's statistic therefore evaluates the local quotient-affine quadratic form directly. Moreover, for two independent Gaussian samples with a common population correlation matrix, the squared geodesic distance, scaled by effective sample size, converges in distribution to $4χ^2_d$, where $d = p(p-1)/2$. For $p=2$, the result reduces to the two-sample Fisher $z$ test.
