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An asymptotically optimal transform of Pearson's correlation statistic

Abstract

It is shown that for any correlation-parametrized model of dependence and any given significance level α(0,1)\alpha\in(0,1), there is an asymptotically optimal transform of Pearson's correlation statistic RR, for which the generally leading error term for the normal approximation vanishes for all values ρ(1,1)\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 RR turns out to be asymptotically optimal for a rather unusual significance level α0.240\alpha\approx0.240, whereas Fisher's transform RFR_F of RR is asymptotically optimal for the limit significance level α=0\alpha=0. In the SquareV model, Pearson's RR is asymptotically optimal for a still rather high significance level α0.159\alpha\approx0.159, whereas Fisher's transform RFR_F of RR is not asymptotically optimal for any α[0,1]\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 α\alpha is in fact asymptotically better than RR and RFR_F in wide ranges of values of the significance level, including α\alpha itself. Extensive computer simulations for the BVN and SquareV models of dependence are presented, which suggest that, for sample sizes n100n\ge100 and significance levels α{0.01,0.05}\alpha\in\{0.01,0.05\}, the mentioned asymptotically optimal transform of RR generally outperforms both Pearson's RR and Fisher's transform RFR_F of RR, the latter appearing generally much inferior to both RR and the asymptotically optimal transform of RR in the SquareV model.

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