Variance-stabilizing and Confidence-stabilizing Transformations for the Normal Correlation Coefficient with Known Variances

Fosdick and Raftery (2012) revisited the classical problem of inference for a bivariate normal correlation coefficient ρ when the variances are known. They considered several frequentist and Bayesian estimators, the former including the maximum likelihood estimator (MLE), but did not obtain the standard errors of these estimators or confidence intervals for ρ. Here we present a new variance-stabilizing transformation y for the MLE in the known-variance case. Adjusting y appropriately according to the sample size n produces a “confidence-stabilizing” transformation yn that provides more accurate interval estimates for ρ than the MLE, as does Fisher’s classical z transformation for the MLE in the unknown-variance case. Interestingly, the z transform applied to the MLE for the unknown-but-equal-variance case performs well in the known-variance case for smaller values of ρ. Both these methods are also useful for comparing two or more correlation coefficients in the known-variance case; hypothesis testing in this case is also discussed.