# Why people often optimize the determinant of $(X'\Sigma X)^{-1}$

Say I have a random vector $Y\sim N(X\beta,\Sigma)$ and $\Sigma\neq\sigma^2 I$. That is, the elements of $Y$ (given $X\beta$) are correlated.

The natural estimator of $\beta$ is $(X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}Y$, and $\text{var}(\hat{\beta})=(X'\Sigma^{-1}X)^{-1}$

In a design context, the experimenter can fiddle with the design which will result in different $X$ and $\Sigma$ thus different $\text{var}(\hat{\beta})$. To choose an optimal design, I see that people often try to minimizes the determinant of $(X'\Sigma^{-1} X)^{-1}$, what is the intuition behind this?

Why not, say, minimizes the sum of its elements?

As a design criterion, to minimize the determinant of $(X'\Sigma^{-1} X)^{-1}$, which is the same as maximizing the determinant of $(X'\Sigma^{-1} X)$, is known as D-optimal experimental design. The determinant of a covariance matrix is known as the generalized variance, so we are minimizing the generalized variance. Other functionals of the covariance matrix could be used as a criterion, but what you propose (minimizing sum of its elements) does not make much sense. The D-optimality criterion has the big practical advantage of being invariant under linear transformations of the regressor variables, which is a big practical advantage. Invariance means that the optimality is not influenced by such things as choice of measurements units, (such as m or k m). With non-invariant optimality criteria the result could depend on such irrelevant things as choice of measurement units.