Prediction in the linear model: A direct approach

Abstract.

In the present paper we propose a direct approach to prediction in the linear model E[Y]=Xβ, where some but not all of the coordinates of Y are observable, X is a known design matrix, β is an unknown parameter vector, and Var[Y] is known and positive definite. To construct a predictor Y ^* ₂ of the non-observable part Y ₂ of Y such that E[Y ^* ₂]=E[Y ₂] and Y ^* ₂=B ^* Y ₁, where B ^* is a matrix and Y ₁ is the observable part of Y, we minimize the expected weighted loss

over all estimators of β which are unbiased and linear in Y ₁. The predictor Y ^* ₂ is then obtained from a canonical decomposition of the expected loss into an approximation part for Y ₁ and a prediction part for Y ₂.