Using U Statistics to Derive the Asymptotic Distribution of Fisher's Z Statistic

Abstract A simple derivation of the asymptotic distribution of Fisher's Z statistic for general bivariate parent distributions F is obtained using U-statistic theory. This method easily reveals that the asymptotic variance of Z generally depends on the correlation ρ and on certain moments of F. It also reveals the particular structure of F that makes the asymptotic variance of Z independent of ρ, and shows that there are many distributions F with this property. The bivariate normal is only one such F.