Best Linear Unbiased Prediction

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 E ( u ) = 0 , var ( u ) = σ 2 h , and cov ( y , u ) = σ 2 k . $$\displaystyle \mbox{E}(u) = 0, \quad \mbox{var}(u) = \sigma ^2h, \quad \mbox{and cov}(\mathbf {y}, u) = \sigma ^2\mathbf {k}. $$ Here h is a specified nonnegative scalar and k is a specified n-vector such that the matrix W k k T h $$\left (\begin {array}{cc}\mathbf {W} & \mathbf {k}\\{\mathbf {k}}^T & h \end {array}\right )$$ ,which is equal to (1∕σ 2) times the variance–covariance matrix of y u $$\left (\begin {array}{c}\mathbf {y}\\u\end {array}\right )$$ , 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, Aitken model prediction-extended and to the inference problem as the general prediction problem Prediction (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 .