Best Linear Unbiased Prediction

Abstract

Suppose, as in Chap. 11, that the model for y is an Aitken model. In this chapter, however, rather than considering the problem of estimating c ^Tβ under that model (which we have already dealt with), we consider the problem of estimating or, to state it more accurately, predictingτ ≡c ^Tβ + u, where u is a random variable satisfying

$$\displaystyle \mbox{E}(u) = 0, \quad \mbox{var}(u) = \sigma ^2h, \quad \mbox{and cov}(\mathbf {y}, u) = \sigma ^2\mathbf {k}. $$

Here h is a specified nonnegative scalar and k is a specified n-vector such that the matrix $\left (\begin {array}{cc}\mathbf {W} & \mathbf {k}\\{\mathbf {k}}^T & h \end {array}\right )$,which is equal to (1∕σ ²) times the variance–covariance matrix of $\left (\begin {array}{c}\mathbf {y}\\u\end {array}\right )$, is nonnegative definite. We speak of “predicting τ” rather than “estimating τ” because τ is now a random variable rather than a parametric function (although one of the summands in its definition, namely c ^Tβ, is a parametric function). We refer to the joint model for y and u just described as the prediction-extended Aitken model, and to the inference problem as the general prediction problem (under that model). In the degenerate case in which u = 0 with probability one, the general prediction problem reduces to the estimation problem considered in Chap. 11.