It is shown that for any correlation-parametrized model of dependence and any given significance level α ∈ (0, 1), there is an asymptotically optimal transform of Pearson’s correlation statistic R, for which the generally leading error term for the normal approximation vanishes for all values ρ ∈ (−1, 1) of the correlation coefficient. This general result is then applied to the bivariate normal (BVN) model of dependence and to what is referred to in this paper as the SquareV model. In the BVN model, Pearson’s R turns out to be asymptotically optimal for a rather unusual significance level α ≈ 0.240, whereas Fisher’s transform R_F of R is asymptotically optimal for the limit significance level α = 0. In the SquareV model, Pearson’s R is asymptotically optimal for a still rather high significance level α ≈ 0.159, whereas Fisher’s transform R_F of R is not asymptotically optimal for any α ∈ [0, 1]. Moreover, it is shown that in both the BVN model and the SquareV model, the transform optimal for a given value of α is in fact asymptotically better than R and R_F in wide ranges of values of the significance level, including α itself. Extensive computer simulations for the BVN and SquareV models of dependence suggest that, for sample sizes n ≥ 100 and significance levels α ∈ {0.01, 0.05}, the mentioned asymptotically optimal transform of R generally outperforms both Pearson’s R and Fisher’s transform R_F of R, the latter appearing generally much inferior to both R and the asymptotically optimal transform of R in the SquareV model.

结果表明，对于任何相关参数化的依赖模型和任何给定的显着性水平α ε (0, 1)，皮尔逊相关统计量R存在渐近最优变换，其中正态近似的一般领先误差项消失相关系数的所有值ρ ∈ (−1, 1)。然后将此一般结果应用于二元正态 (BVN) 相关模型以及本文中称为 SquareV 模型的模型。在 BVN 模型中，Pearson 的R对于相当不寻常的显着性水平α ≈ 0.240 是渐近最优的，而R的Fisher 变换R _F对于极限显着性水平α = 0 是渐近最优的。在 SquareV 模型中，Pearson 的R是对于仍然相当高的显着性水平α ≈ 0.159 是渐近最优的，而R的Fisher 变换R _F对于任何α ∈ [0, 1]都不是渐近最优的。此外，结果表明，在 BVN 模型和 SquareV 模型中，给定α值的最佳变换实际上在显着性水平的广泛值范围（包括α本身）中渐近优于R和R _F。对 BVN 和 SquareV 依赖模型的广泛计算机模拟表明，对于样本大小n ≥ 100 和显着性水平α ∈ {0.01, 0.05 } ，上述R的渐近最优变换通常优于 Pearson R和 Fisher 变换R _F，后者通常远不如SquareV 模型中的R和R的渐近最优变换。