Moments of the Nonnegative Adjusted Estimator of Squared Multiple Correlation

I present the moments of the nonnegative adjusted estimator of the squared multiple correlation $\rho 2,$ the coefficient of determination for random-predictor regression. This estimator, first proposed by Ezekiel, replaces with zero the negative estimates from the well-known adjusted estimator proposed by Fisher that, in turn, corrects the positive bias of the sample R². Although Fisher’s version is presented in texts, Ezekiel’s version is used in practice. Each moment comprises a binomial sum of a negative binomial series of incomplete beta functions. Numerical computations, for which an R function is provided, are required to examine the moments. Ezekiel’s estimator is positively biased for smaller $\rho 2$ and negatively biased for larger. It dominates Fisher’s via MSE. It does not dominate R², but the MSE of Ezekiel’s estimator can be substantially smaller but at most negligibly larger. Possible applications to powers of $\rho 2$ and to other adjusted estimators are briefly discussed.