Kiefer–Wolfowitz equivalence theorem

Remarks

For every design with nonsingular information matrix, $\int d(x, \xi) \, d\xi(x) = \tr(M(\xi)^{-1} M(\xi)) = p$. So $\max_x d(x, \xi) \ge p$ always holds, and this explains the number $p$ in condition 3. The function $\xi \mapsto \log \det M(\xi)$ is concave on the convex set of designs, and its directional derivative at $\xi$ towards the point mass $\delta_x$ equals $d(x, \xi) - p$. So condition 3 says that no direction increases $\log \det M$, which for a concave function means a maximum. Example: for the line $f(x) = (1, x)^T$ on $\mathcal{X} = [-1, 1]$, the design $\xi^* = \frac12 \delta_{-1} + \frac12 \delta_{1}$ has $M(\xi^*) = I_2$. Then $d(x, \xi^*) = 1 + x^2 \le 2 = p$, so $\xi^*$ is D-optimal and G-optimal. The theorem was proved by Jack Kiefer and Jacob Wolfowitz in 1960.

Going deeper

For a student

The equivalence theorem turns a hard optimization over probability measures into a checkable condition: compute $d(x, \xi)$ and see whether it stays below $p$. It is the basis of the Fedorov–Wynn type algorithms, which add weight at the point where $d(x, \xi)$ is largest. Similar equivalence theorems hold for other concave criteria of optimal experimental design, with the variance function replaced by the corresponding directional derivative.