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 $4\chi^2_d$, where $d = p(p-1)/2$. For $p=2$, the result reduces to the two-sample Fisher $z$ test.
Comparing correlation matrices across time or stress scenarios is critical in quantitative finance and multivariate statistics, yet sample estimation noise often obscures whether an observed distance reflects a true structural shift. We derive the asymptotic sampling distribution of the intrinsic off-log (log-Euclidean) distance between two independently estimated full-rank correlation matrices under the null hypothesis that their population correlation matrices coincide. Under general sampling with finite fourth moments, the scaled squared distance converges to a weighted sum of independent $\chi_1^2$ variables, with weights determined by the asymptotic covariance of the Generalized Fisher Transformation (GFT) coordinates. Under Gaussian sampling at independence, this simplifies to a parameter-free $4\chi_d^2$ law. To calibrate tail probabilities, we provide closed-form cumulant generating functions, Lugannani--Rice saddlepoint quantiles, and an explicit Chernoff envelope requiring no root-finding. The first moment of the limiting law establishes a simple rule of thumb for the baseline expected distance under the null hypothesis ($\operatorname E[d_{\mathrm{LE}}] \lesssim 2\sqrt{d/n}$ near independence), quantifying the average separation induced strictly by estimation error. We establish plug-in consistency, present an explicit Gaussian covariance factorization, compare the distance statistic with coordinate Wald tests, and characterize its local power.
A. Kuketayev· 0 citations
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