Improvements on removing non-optimal support points in D-optimum design algorithms

We improve the inequality used in Pronzato [2003. Removing non-optimal support points in D-optimum design algorithms. Statist. Probab. Lett. 63, 223-228] to remove points from the design space during the search for a $D$-optimum design. Let $\xi$ be any design on a compact space $$\mathcal{X}$ \subset $\mathbb{R}$^m$withanonsingularinformationmatrix,andlet$m+\epsilon$bethemaximumofthevariancefunction$d(\xi,$\mathbf{x}$)$ over all $\mathbf{x}\in \mathcal{X}$. We prove that any support point ${\mathbf{x}}_{\ast }$ of a $D$-optimum design on $\mathcal{X}$ must satisfy the inequality $d(\xi,$\mathbf{x}$_{*}) \geq m(1+\epsilon/2-\sqrt{\epsilon(4+\epsilon-4/m)}/2)$. We show that this new lower bound on $d(\xi ,{\mathbf{x}}_{\ast })$ is, in a sense, the best possible, and how it can be used to accelerate algorithms for $D$-optimum design.