Comparisons of Alternative Predictors under the Balanced One-Way Random Model

Prediction of an arbitrary linear combination of the random effects of a balanced one-way random model is investigated. Alternative two-stage predictors are compared on the basis of their conditional (on the random effects) and unconditional bias and mean squared errors. When the true value of the ratio of expected mean squares is known, there exists a best linear unbiased predictor (BLUP). When the true value is unknown, a two-stage predictor, obtained from the BLUP by replacing the true value with an estimated value, can be used. When the ratio of expected mean squares is estimated by maximum likelihood, Bayesian methods, or various related methods, a two-stage predictor is obtained whose properties compare favorably with, for example, those of the least squares predictor and the positive-part James-Stein predictor.