BLUP (Best Linear Unbiased Prediction) and Beyond

TLDR: In this article, the authors considered the problem of predicting the value of an unobservable random variable w from the values of an observable random vector y under each of four states of knowledge about the joint distribution of w and y, ranging from complete knowledge to "no" knowledge.

Abstract: The problem considered is that of predicting the value of an unobservable random variable w from the value of an observable random vector y. This problem is considered under each of four states of knowledge about the joint distribution of w and y, ranging from complete knowledge to “no” knowledge. Point predictors, estimators of the mean squared error of prediction, and interval predictors are presented for each case. Both frequentist and Bayesian approaches are discussed and relationships between the two are pointed out. Specifics are given for the prediction of a linear combination of the fixed and random effects in a mixed linear model. The results are illustrated by applying them to some animal breeding data.