ABSTRACT
As pointed out in Section 1.1, for any design ξN the LSE θˆN is the best one in the sense that
Var[θˆN ] = min ˜θ
Var[θ˜N ], (2.1)
where minimization is performed with respect to all linear unbiased estimators. In a more general nonlinear situation, as discussed in Section 1.5, the MLE θN also possesses certain optimality properties in that asymptotically, as N → ∞, the variance-covariance matrix Var[θN ] is inversely proportional to M−1(ξN ,θ) and attains the lower bound in Crame`r-Rao inequality under certain regularity conditions on the distribution of {yij}; see references after (1.89). So the next rather natural question is whether it is possible to
“minimize” Var[θˆN ] or Var[θN ] among all possible designs. As in Chapter 1, we start with the simpler linear model (1.1), (1.2) for
which various optimization problems (criteria of optimality) are introduced. Then in Section 2.2.3 we extend the criteria to the more general setting.
