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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 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 RF 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 RF 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 RF 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 RF of R, the latter appearing generally much inferior to both R and the asymptotically optimal transform of R in the SquareV model.

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  1. R. N. Bhattacharya and J. K. Ghosh, “On the Validity of the formal Edgeworth expansion”, Ann. Statist. 6, 434–451 (1978).

    MathSciNet  Article  Google Scholar 

  2. L. H. Y. Chen and Q. M. Shao, “Normal Approximation for Nonlinear Statistics Using a Concentration Inequality Approach”, Bernoulli 13, 581–599 (2007).

    MathSciNet  Article  Google Scholar 

  3. D. J. G. Farlie, “The Performance of Some Correlation Coefficients for a General Bivariate Distribution”, Biometrika 47, 307–323 (1960).

    MathSciNet  Article  Google Scholar 

  4. R. A. Fisher, “Frequency Distribution of the Values of the Correlation Coefficient in Samples from an Indefinitely Large Population”, Biometrika 10, 507–521 (1915).

    Google Scholar 

  5. A. K. Gayen, “The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes”, Biometrika 38, 219–247 (1951).

    MathSciNet  Article  Google Scholar 

  6. H. Hotelling, “New Light on the Correlation Coefficient and Its Transforms”, J. Roy. Statist. Soc. Ser. B. 15, 193–225; discussion, 225–232 (1953).

    MathSciNet  MATH  Google Scholar 

  7. R. B. Nelsen, An Introduction to Copulas, in Springer Series in Statistics (Springer, New York, 2006), 2nd ed..

    Google Scholar 

  8. I. Pinelis, “On l’Hospital-Type Rules for Monotonicity”, J. Inequal. Pure Appl. Math. JIPAM 7, (2005), Article 40, 19 pp. (electronic),

  9. I. Pinelis, “An Asymptotically Optimal Transform of Pearson’s Correlation Statistic”, (20019).

  10. I. Pinelis and R. Molzon, “Optimal-Order Bounds on the Rate of Convergence to Normality in the Multivariate Delta Method”, Electron. J. Statist. 10, 1001–1063 (2016).

    MathSciNet  Article  Google Scholar 

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Correspondence to I. Pinelis.

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Pinelis, I. An Asymptotically Optimal Transform of Pearson’s Correlation Statistic. Math. Meth. Stat. 28, 307–318 (2019).

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  • hypothesis testing
  • Pearson’s correlation statistic
  • Fisher’s z transform
  • asymptotically optimal transform
  • models of dependence
  • copulas
  • bivariate normal distribution

AMS 2010 Subject Classification

  • 62E20
  • 62F03
  • 62F12