On the distribution of Fisher's transformation of the correlation coefficient

Abstract

As the sample size increases, the coefficient of skewness of the Fisher's transformation, z = (1/2) log ((l+r)/(l-r)), of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the usual normal approximation for its distribution can be improved by adjusting for the excess of its kurtosis. This is accomplished by mixing the approximating normal distribution with a logistic distribution. The resulting mixture approximation which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1973).

Keywords:

• approximations
• correlation coefficient
• Fisher's transformation
• logistic distribution
• mixtures