Combined estimators for the correlation coefficient

Three combined estimators for the bivariate normal correlation parameter are considered. The data consist of k independent sample correlation coefficients and it is assumed that the underlying correlation parameters are all equal to ρ. Based upon the joint density function of the sample correlations a combined estimator of ρ is obtained as an approximation to the maximum likelihood solution. Two linearly combined estimators are also considered. One of them is based on Fisher's z-transformation of the sample correlations and the other on an unbiased estimator of ρ. The comparison of these three estimators indicates that the combined (approximate) MLE has a slightly smaller estimated mean squared error relative to the other two combined methods of estimation, but it does so at the expense of a relatively larger bias.