Unbiased Estimation of Certain Correlation Coefficients

Abstract

This paper deals with the unbiased estimation of the correlation of two variates having a bivariate normal distribution (Sec. 2), and of the intraclass correlation, i.e., the common correlation coefficient of a $p$-variate normal distribution with equal variances and equal covariances (Sec. 3). In both cases, the estimator has the following properties. It is a function of a complete sufficient statistic and is therefore the unique (except for sets of probability zero) minimum variance unbiased estimator. Its range is the region of possible values of the estimated quantity. It is a strictly increasing function of the usual estimator differing from it only by terms of order $1/n$ and consequently having the same asymptotic distribution. Since the unbiased estimators are cumbersome in form in that they are expressed as series or integrals, tables are included giving the unbiased estimators as functions of the usual estimators. In Sec. 4 we give an unbiased estimator of the squared multiple correlation. It has the properties mentioned in the second paragraph except that it may be negative, which the squared multiple correlation cannot. In each case the estimator is obtained by inverting a Laplace transform. We are grateful to W. H. Kruskal and L. J. Savage for very helpful comments and suggestions, and to R. R. Blough for his able computations.