Exercise 10.9 From inspecting Figure 10.2, give the least squares CLUE for Example 10.2.2. Do not do any matrix manipulations.

Remark Suppose that we are analyzing the model Y = Xβ +

e, E

(e) = 0, Cov

(e) = Σ(θ), where Σ(θ) is some nonnegative definite matrix depending on a vector of unknown parameters θ. The special case where Σ(θ) = σ2V is what we have been considering so far. It is clear that if θ is known, our current theory gives BLUEs. If it happens to be the case that for any value of θ the BLUEs are identical, then the BLUEs are known even though θ may not be. This is precisely what we have been doing with Σ(θ) = σ2V. We have found BLUEs for any value of σ2, and they do not depend on σ2. Another important example of this occurs when C(Σ(θ)X) ⊂ C(X) for any θ. In this case, least squares estimates are BLUEs for any θ, and least squares estimates do not depend on θ, so it does not matter that θ is unknown. The split plot design model is one in which the covariance matrix depends on two parameters, but for any value of those parameters, least squares estimates are BLUEs. Such models are examined in the next chapter.