We study the finite-sample behavior of the Generalized Fisher Transformation (GFT), the parametrization of a correlation matrix $C$ by $γ(C)=\operatorname{vecl}\log C$. The GFT coordinates extend Fisher's transformation to dimension $n>2$: for elliptical data their finite-sample distributions are close to Gaussian. More strikingly, the coordinates are nearly uncorrelated and their covariance is largely invariant to $C$. This approximate orthogonality and invariance make GFT-based inference far better behaved in finite samples than inference based on sample correlations or element-wise Fisher transformed correlations, yielding estimation errors that are approximately Gaussian, weakly dependent, and nearly pivotal.

翻译：我们研究了广义Fisher变换（GFT）在有限样本下的行为，该变换通过$γ(C)=\operatorname{vecl}\log C$对相关矩阵$C$进行参数化。GFT坐标将Fisher变换推广至维度$n>2$：对于椭圆分布数据，其有限样本分布接近高斯分布。更引人注目的是，这些坐标几乎不相关，且其协方差在很大程度上不依赖于$C$。这种近似正交性和不变性使得基于GFT的推断在有限样本中远优于基于样本相关性或逐元素Fisher变换相关性的推断，其估计误差近似为高斯分布、弱相关且近乎枢轴量。