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

Abstract

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 $\mathcal{X}\subset {\mathbb{R}}^m$ with a nonsingular information matrix, and let $m+\epsilon$ be the maximum of the variance function $d(\xi ,\mathbf{x})$ 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 $d(\xi ,{\mathbf{x}}_{\ast })\ge m(1+\epsilon /2-\sqrt{\epsilon (4+\epsilon -4/m)}/2)$. 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.

5 pages Statistics and Probability letters available online at: http://www.elsevier.com/locate/stapro

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