An optimal property of least squares weights in prediction models

In predicting scores fromp > 1 observed scores in a sample of sizeñ, the optimal strategy (minimum expected loss), under certain assumptions, is shown to be based upon the least squares regression weights computed from a previous sample. Letting represent the correlation between and the predicted values, and letting represent the correlation between and a different set of predicted values, where w is any weighting system which is not a function of, it is shown that the probability of being less than cannot exceed .50. The relationship of this result to previous research and practical implications are discussed.