The sample squared multiple correlation coefficient is widely used for describing the usefulness of a multiple linear regression model in many areas of science. In this article, the author considers the problem of estimating the squared multiple correlation coefficient and the squared cross-validity coefficient under the assumption that the response and predictor variables have a joint multinormal distribution. Detailed numerical investigations are conducted to assess the exact bias and mean square error of the proposed modifications of established estimators. Notably, the positive-part Pratt estimator and the synthesis of Browne and positive-part Pratt estimators are recommended in the estimation of squared multiple correlation coefficient and squared cross-validity coefficient, respectively, for their overall advantages of incurring the least amount of statistical discrepancy and computational requirement.
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