An asymptotically optimal transform of Pearson's correlation statistic

It is shown that for any correlation-parametrized model of dependence and any given significance level $\alpha\in(0,1)$, 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 $\rho\in(-1,1)$ 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 $\alpha\approx0.240$, whereas Fisher's transform $R_F$ of $R$ is asymptotically optimal for the limit significance level $\alpha=0$. In the SquareV model, Pearson's $R$ is asymptotically optimal for a still rather high significance level $\alpha\approx0.159$, whereas Fisher's transform $R_F$ of $R$ is not asymptotically optimal for any $\alpha\in[0,1]$. Moreover, it is shown that in both the BVN model and the SquareV model, the transform optimal for a given value of $\alpha$ is in fact asymptotically better than $R$ and $R_F$ in wide ranges of values of the significance level, including $\alpha$ itself. Extensive computer simulations for the BVN and SquareV models of dependence are presented, which suggest that, for sample sizes $n\ge100$ and significance levels $\alpha\in\{0.01,0.05\}$, 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.