3.1 D-Optimality Introduction

The first measure of optimality we shall consider is \(D\)-optimality. This is where we seek to maximise the determinant of the Fisher information matrix, \(\mathbf{I}=\mathbf{F}^T \mathbf{F}\).

If you recall from our section on regression, we claimed that the variance of an estimator is a measure of its precision. We also established a formula for the variance of the regression coefficients, \(\boldsymbol{\hat{\beta}}\) using the design matrix, \(\mathbf{F}\):

\[\text{Var}(\boldsymbol{\hat{\beta}}) = \sigma^2 \left(\mathbf{F}^T \mathbf{F}\right)^{-1}\]

where the design matrix is given by

\[\mathbf{F} = \begin{pmatrix} 1 & x_1 \\ 1 & x_2 \\ 1 & x_3 \\ \vdots & \vdots \\ 1 & x_n \end{pmatrix}\]

Therefore, our Fisher information matrix becomes:

\[\begin{align} \mathbf{I} &= \mathbf{F}^T \mathbf{F} \\ \\ & = \begin{pmatrix} 1 & 1 & 1 & \ldots & 1\\ x_1 & x_2 & x_3 & \ldots & x_n \end{pmatrix} \begin{pmatrix} 1 & x_1 \\ 1 & x_2 \\ 1 & x_3 \\ \vdots & \vdots \\ 1 & x_n \end{pmatrix} \\ \\ &= \begin{pmatrix} n & \sum_{i=1}^{n} x_i \\ \sum_{i=1}^{n} x_i & \sum_{i=1}^{n} x_i^2 \end{pmatrix} \end{align}\]

From here, we can compute the determinant of the information matrix:

\[\begin{align} \left|\mathbf{I}\right| &= \begin{vmatrix} n & \sum_{i=1}^{n} x_i \\ \sum_{i=1}^{n} x_i & \sum_{i=1}^{n} x_i^2 \end{vmatrix} \\ \\ &= n \cdot \sum_{i=1}^{n} x_i^2 - \left( \sum_{i=1}^{n} x_i \right)^2 \end{align}\]

This tells us that, for \(D\)-optimality, the precision of the estimators very much depends on your choice of design points, \(\left\{x_i\right\}\), and the number of design points in your experiment, \(n\).