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An asymptotically optimal transform of Pearson's correlation statistic

26 July 2019
I. Pinelis
ArXiv (abs)PDFHTML
Abstract

It is shown that for any correlation-parametrized model of dependence and any given significance level α∈(0,1)\alpha\in(0,1)α∈(0,1), there is an asymptotically optimal transform of Pearson's correlation statistic RRR, for which the generally leading error term for the normal approximation vanishes for all values ρ∈(−1,1)\rho\in(-1,1)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 RRR turns out to be asymptotically optimal for a rather unusual significance level α≈0.240\alpha\approx0.240α≈0.240, whereas Fisher's transform RFR_FRF​ of RRR is asymptotically optimal for the limit significance level α=0\alpha=0α=0. In the SquareV model, Pearson's RRR is asymptotically optimal for a still rather high significance level α≈0.159\alpha\approx0.159α≈0.159, whereas Fisher's transform RFR_FRF​ of RRR is not asymptotically optimal for any α∈[0,1]\alpha\in[0,1]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\alphaα is in fact asymptotically better than RRR and RFR_FRF​ in wide ranges of values of the significance level, including α\alphaα itself. Extensive computer simulations for the BVN and SquareV models of dependence are presented, which suggest that, for sample sizes n≥100n\ge100n≥100 and significance levels α∈{0.01,0.05}\alpha\in\{0.01,0.05\}α∈{0.01,0.05}, the mentioned asymptotically optimal transform of RRR generally outperforms both Pearson's RRR and Fisher's transform RFR_FRF​ of RRR, the latter appearing generally much inferior to both RRR and the asymptotically optimal transform of RRR in the SquareV model.

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