Moments of the Nonnegative Adjusted Estimator of Squared Multiple Correlation

I present the moments of the nonnegative adjusted estimator of the squared multiple correlation (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) and to other adjusted estimators are briefly discussed.