Optimal properties of the conditional mean as a selection criterion.

Rules for selection that maximize the expected merit of selected candidates are discussed. When the proportion selected is constant, selection based on conditional means of merit given the observations is optimum in the above sense, regardless of the distribution. This does not hold if the proportion selected is random. When the expected value of the observations is a linear function of a set of unknown parameters, selection can be based on a vector of "corrected" records, w. It is shown that under normality, the conditional mean of merit given w is the best linear unbiased predictor (BLUP), provided that the expected value of the merit function is the same in all candidates. A Bayesian argument is given to justify the use of BLUP as a selection rule when the expected merit differs from candidate to candidate.