The Fisher Transform of the Pearson Product Moment Correlation Coefficient and Its Square: Cumulants, Moments, and Applications

This paper extends results on the distribution of the Fisher transform of the correlation coefficient (Fisher, 1921). Approaches to obtain exact moments of the Fisher transform for both null and non-null correlations are presented. We extend the classic series expansion formulae of Hotelling (1953) for the moments of the Fisher transform. These results are considered in the context of quadratic functions of the Fisher transform. Some applications of these results are discussed in the context of correlational hypothesis tests and confidence intervals, and a Monte Carlo experiment is used to demonstrate how application of these results impact the small sample performance of select tests on correlations.