Decomposition of Prediction Error

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. A (point) predictor or predictors are presented for each case. Prediction error is decomposed so that each component reflects an absence of information. Under certain conditions, these components are uncorrelated and have zero means. An exact or approximate expression is given for the variance of each component. Key Words: Best linear unbiased predictionMixed linear modelsEmpirical Bayes estimationJames—Stein estimatorVariance components