Estimating realized random effects

Random sampling of a finite population of subjects leads to observed values on the selected subjects. When the measure on a subject includes response error, the observed response for a selected subject can be distinguished from the latent value representing the subject's parameter. We derive linear unbiased minimum MSE estimates of a realized subject's latent value from sample data, where these data are considered to have arisen from three possible sample spaces: a space conditional on the realized sample subjects; a space conditional on a single realized subject in the sample; and a space unconditional on the realized sample subjects. The derivation results in the ordinary mean estimate (BLUE) for the first two sample spaces, and the best linear unconditionally unbiased estimate (BLUUE) commonly referred to as the best linear unbiased predictor (BLUP) for the third sample space. Focusing on estimates of a single realized subject's latent value and the second sample space, we identify a range of latent parameter values where BLUUE estimates have smaller MSE than BLUE estimates. The results offer a direct frequentist development of BLUE and BLUUE estimates, providing clear interpretation of differences when estimating a realized subject's latent value. Examples are given, along with a strategy for practically implementing a choice between estimators in a simple setting (assuming normality). The results make use of distributional assumptions only when choosing between estimators based on an observed response.